operators in sobolev morrey spaces -...

Sede Amministrativa: Universita degli Studi di Padova

SCUOLA DI DOTTORATO DI RICERCA IN: Scienze

Matematiche

INDIRIZZO: Matematica

CICLO: XXIV

Operators in Sobolev

Morrey spaces

Direttore della Scuola: Ch.mo Prof. Paolo Dai Pra

Coordinatore d’indirizzo: Ch.mo Prof. Franco Cardin

Supervisore: Ch.mo Prof. Victor I. Burenkov

Ch.mo Prof. Massimo Lanza de Cristoforis

Dottoranda: Nurgul Kydyrmina

To my Father and Mother

i

Abstract

Morrey spaces were introduced by Charles Morrey in 1938. They are a useful

tool in the regularity theory of partial differential equations, in real analysis

and in mathematical physics.

In the nineties of the XX century an active study of general Morrey-type

spaces characterized by a functional parameter has started to develop. A

number of results on boundedness of classical operators in general Morrey-

type spaces were obtained.

At the beginning of the XXI century there were new active developments in

this area. In the last decade many mathematicians do research on smoothness

spaces related to Morrey spaces. Among these spaces the Sobolev-type spaces

play an important role.

In the thesis Sobolev spaces built on Morrey spaces are studied, which are

also referred to as Sobolev Morrey spaces. These are spaces of functions which

have derivatives up to certain order in Morrey spaces.

We analyze some basic properties of Morrey spaces and of Sobolev Mor-

rey spaces. Then we consider the embedding and multiplication operators

in Sobolev Morrey spaces. Finally, the dissertation provides a study of the

composition operator in Sobolev Morrey spaces.

The results presented in the thesis have been obtained under supervision

of Professors V.I. Burenkov and M. Lanza de Cristoforis.

iii

Sunto

Gli spazi di Morrey sono stati introdotti da Charles Morrey nel 1938. Essi sono

uno strumento utile nella teoria della regolarita per equazioni differenziali alle

derivate parziali, in analisi reale ed in fisica matematica.

Negli anni novanta del XX secolo ha iniziato a svilupparsi un attivo stu-

dio degli spazi di Morrey di tipo generalizzato che sono caratterizzati da un

parametro funzionale. E stato ottenuto un cero numero di risultati sulla limi-

tatezza degli operatori classici negli spazi di Morrey di tipo generalizzato.

All’inizio del XXI secolo ci sono stati nuovi e attivi sviluppi in questa area.

Nell’ultima decade molti matematici hanno svolto ricerche su spazi funzionali

relativi agli spazi di Morrey. Tra questi spazi gli spazi di tipo Sobolev giocano

un ruolo importante.

Nella tesi si studiano Spazi di Sobolev costruiti su spazi di Morrey, anche

detti spazi di Sobolev Morrey. Questi sono spazi di funzioni che hanno derivate

fino ad un certo ordine negli spazi di Morrey.

Si analizzano alcune proprieta di base degli spazi di Morrey e degli spazi

di Sobolev-Morrey. Poi si considerano operatori di immersione e di moltipli-

cazione negli spazi di Sobolev Morrey. La terza parte della tesi presenta uno

studio degli operatori di composizione negli spazi di Sobolev Morrey.

I risultati presentati nella tesi sono stati ottenuti sotto la supervisione dei

Professori V.I. Burenkov and M. Lanza de Cristoforis.

v

Acknowledgements

First and foremost, I would like to express my sincere gratitude to both of

my supervisors Prof. V.I. Burenkov and Prof. M. Lanza de Cristoforis for

the continuous support of my Ph.D study and research, for their patience,

motivation, and enthusiasm. Their guidance helped me in all the time of

research and writing of this thesis. It has been an honor for me to be their

Ph.D. student.

A special gratitude I give to Prof. E.S. Smailov who have always been a

constant source of encouragement and inspiration during my graduate study.

His unconditional support has been essential all these years.

I acknowledge the University of Padova for giving me the opportunity to

study in Italy and for providing financial assistance.

Last but not the least, I would like to thank my family for their unflagging

love and encouragement throughout my life. I thank my parents, Alim and

Bagila, for their faith in me, for their prayers and supporting me whenever I

needed it. I love you all dearly.

vii

Contents

Introduction 1

Notation 5

1 Morrey and Sobolev Morrey spaces 9

1.1 General Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Approximation by C∞ functions in Morrey spaces . . . . . . . . 20

1.3 Preliminaries on integral operators . . . . . . . . . . . . . . . . 28

1.4 Sobolev spaces built on Morrey spaces . . . . . . . . . . . . . . 36

2 The embedding and multiplication operators in Sobolev Mor-

rey spaces 43

2.1 The Sobolev Embedding Theorem . . . . . . . . . . . . . . . . . 43

2.2 Approximation by C∞ functions in Sobolev Morrey spaces . . . 48

2.3 Multiplication Theorems for Sobolev Morrey spaces . . . . . . . 50

3 The composition operator in Sobolev Morrey spaces 61

3.1 Composition operator in Morrey spaces . . . . . . . . . . . . . . 61

3.2 Composition operator in Sobolev Morrey spaces . . . . . . . . . 63

3.3 Continuity of the composition operator in Sobolev Morrey spaces 68

3.4 Lipschitz continuity of the composition operator in Sobolev Mor-

rey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Differentiability properties of the composition operator in Sobolev

Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Bibliography 79

ix

Introduction

This dissertation is devoted to Sobolev spaces built on Morrey spaces, also

referred to as Sobolev Morrey spaces, i.e., to the spaces of functions which

have derivatives up to a certain order in Morrey spaces.

In the first part of the dissertation we analyze some basic properties of

Morrey spaces and of Sobolev Morrey spaces. In particular,

(i) We characterize the functions in a Morrey space which can be approxi-

mated by smooth functions, as the functions which belong to a specific

subspace of the Morrey space, which we call the ‘little’ Morrey space.

(ii) Contrary to the classical Sobolev spaces built on the Lp spaces with

p < ∞, the Sobolev spaces built on Morrey spaces are not separable

spaces even if p <∞ and we cannot expect that the set of C∞ functions

of a Sobolev Morrey space be dense in a Sobolev Morrey space. However,

we show that the functions in a Sobolev space built on little Morrey

spaces can be approximated by C∞ functions.

In the second part of the dissertation we consider the embedding and mul-

tiplication operators in Sobolev Morrey spaces. Namely,

(i) We prove a Sobolev Embedding Theorem for Sobolev Morrey spaces.

The proof is based on the Sobolev Integral Representation Theorem and

on a recent results on Riesz potentials in generalized Morrey spaces of

Burenkov, Gogatishvili, Guliyev, Mustafaev [14] and on estimates on the

Riesz potentials contained in the dissertation. We mention that a Sobolev

Embedding Theorem for Sobolev Morrey spaces had been proved by

Campanato [19, Thm. II.2, p. 75], for a subspace of our Sobolev Morrey

space which corresponds to the closure of the set of smooth functions in

1

Introduction

our Sobolev Morrey space. The methods of the present dissertation are

considerably different from those of Campanato.

(ii) We prove a multiplication Theorem for Sobolev Morrey spaces which ex-

tends to Sobolev Morrey spaces a known result of Zolesio [61] for classical

Sobolev space (see also Valent [58], Runst and Sickel [51]).

Both the Sobolev Imbedding Theorem of (i) and the multiplication Theo-

rem of (ii) have been proved for bounded domains with the cone property. We

believe that one could prove the same type of results for unbounded domains

with the cone property and in particular for the entire space.

Then in the third part of the dissertation, we consider the composition

operator in Sobolev Morrey spaces, and as a first step we do so for Sobolev

Morrey spaces of the first order. Let Ω be a bounded open subset of Rn with

the cone property. Let W 1,λp (Ω) be the Sobolev space of functions with deriva-

tives up to order 1 in the Morrey space Mλp (Ω) with exponents λ ∈ [0, n/p],

p ∈ [1,+∞].

Let Ω1 be a bounded open subset of R. Let W 1,λp (Ω,Ω1) denote the set of

functions of W 1,λp (Ω) which map Ω to Ω1.

Let C0,1(Ω1) denote the space of Lipschitz continuous functions from Ω1

to R. Let r be a natural number. Let Cr(Ω1) denote the space of r times

continuously differentiable functions from Ω1 to R.

Then we prove the following results.

(j) We prove that if f ∈ C0,1(Ω1) and if g ∈ W 1,λp (Ω) has values in Ω1, then

the composite function f g belongs to W 1,λp (Ω) and the norm of f g

can be estimated in terms of the norms of f and of g. We note that in

case λ = 0, which corresponds to a classical Sobolev space such a result

is well known (see Marcus and Mizel [35]).

(jj) We exploit an abstract scheme of Lanza de Cristoforis [30] and prove

that if (1 + λ) > n/p, then the composition map T from Cr+1(Ω1) ×

W 1,λp (Ω,Ω1) which takes a pair (f, g) to the composite function f g is

r-times continuously Frechet differentiable. We note that in case λ = 0

the result of the present dissertation improves a corresponding result of

Valent [58] for case r = 1.

2

Introduction

(jjj) We prove that if f ∈ C1,1loc (R) and if (1 + λ) > n/p, then the map which

takes g to fg is Lipschitz continuous on the bounded subsets ofW 1,λp (Ω).

For a related result in the Besov space setting, we refer to Bourdaud and

Lanza de Cristoforis [9].

We believe that our sufficient conditions on f of (j), (jj), (jjj) are optimal,

just as they have been shown to be optimal in the frame of Sobolev spaces,

which corresponds to case λ = 0 (see Appell and Zabreiko [4, Ch. 9], Runst

and Sickel [51, Ch. 5], Bourdaud and Lanza de Cristoforis [9].)

We believe that by proving the Sobolev Imbedding Theorem of (i) and the

multiplication Theorem of (ii) above for unbounded domains with the cone

property, one could prove also the results of (j)–(jjj) for unbounded domains

with the cone property and in particular for Ω = Rn.

The composition operator has been considered by several authors. For

extensive references, we refer to the monographs of Appell and Zabreiko [4,

Ch. 9], of Runst and Sickel [51], of Dudley and Norvaisa [23], and to the

recent survey paper Bourdaud and Sickel [11]. In particular, the continuity,

the Lipschitz continuity and the higher order differentiability of f g has a

function of both f and g has long been investigated.

In the Sobolev space setting, we mention in particular Marcus and

Mizel [34]–[40], Adams [3], Szigeti [56], [57], Valent [58], [59], Gol’dshtein and

Reshetnyak [26], Drabek and Runst [22], Musina [41], Bourdaud and Meyer

[10], Bourdaud [6], [7], Bourdaud and Kateb [8], Sickel [53]. As far as con-

sidering the differentiability of the composition operator when both the func-

tions f and g belong to a Sobolev space, we mention a paper of Brokate and

Colonuis [12], and of Lanza de Cristoforis [32].

The results of this dissertation will appear as joint work with the supervi-

sors V.I. Burenkov and M. Lanza de Cristoforis.

3

Notation

N denotes the set of all natural numbers including 0. Throughout the paper,

n is an element of N \ 0.

As usual, R is the set of all real numbers, Rn is the n-dimensional Euclidean

space, and

Nn = N× · · · × N︸︷︷︸n

is the set of multi-indices.

P(Rn) – the linear space of polynomials with real coefficients and n real

variables.

B(x, r) – the open ball of radius r > 0 centered at the point x ∈ Rn.

vn – the volume of the unit ball in Rn.

supp f – the support of a function f .

f←(D) – f -preimage of a set D.

Dα ≡ ∂α1+···+αnf

∂xα11 ... ∂xαn

n– the (ordinary) derivative of the function f of order α,

and

Dαw ≡

(∂α1+···+αnf

∂xα11 ... ∂xαn

n

)w– the weak derivative of the function f of order α.

For an arbitrary nonempty set Ω ⊂ Rn we shall denote by:

diamΩ – the diameter of Ω,

Ω or cl(Ω) – the closure of Ω,

χΩ – the characteristic function of Ω, i.e. χΩ(ξ) = 1 if ξ ∈ Ω and χΩ(ξ) = 0

if ξ ∈ Rn \ Ω,

C0(Ω) – the space of functions continuous on Ω,

C0b (Ω) – the Banach space of functions continuous and bounded on Ω with

the sup norm in Ω,

C0ub(Ω) – the Banach space of uniformly continuous and bounded functions

5

Notation

on Ω with the sup norm in Ω,

Cm(Ω) (m ∈ N) – the Banach space of m-times continuously differentiable

functions on Ω,

C∞(Ω) =∩

m∈NCm(Ω) – the space of infinitely continuously differentiable

functions on Ω,

C∞c (Ω) – the space of functions in C∞(Ω) with compact support.

For a measurable nonempty set Ω ⊂ Rn we shall denote by:

Lp(Ω) (1 ≤ p < ∞) – the Banach space of functions f measurable on Ω

such that the norm

∥f∥Lp(Ω) =

∫Ω

|f(x)|pdx

1p

<∞.

L∞(Ω) – the Banach space of functions f measurable on Ω such that the

norm

∥f∥L∞(Ω) = ess supx∈Ω

|f(x)| <∞.

For an open nonempty set Ω ⊂ Rn we shall denote by:

Llocp (Ω) (1 ≤ p ≤ ∞) – the set of functions f defined on Ω such that for

each compact K ⊂ Ω f ∈ Lp(K),

B(Ω) ≡ f : Ω → R : f is bounded,

M(Ω) ≡ f : Ω → R : f is measurable,

M(Ω) – the factor spaceM(Ω)/Θ(Ω), where Θ(Ω) is the set of all functions

defined on Ω which are equal to 0 almost everywhere on Ω,

Cm(Ω) – the subspace of Cm(Ω) of functions f such that f and its deriva-

tives Dαf of order |α| ≤ m can be extended with continuity to Ω,

C∞(Ω) – the set of functions f from Ω to R such that there exist an open

neighborhood U of Ω and a function F ∈ C∞(U) such that the restriction of

F to Ω coincides with f .

Definition 0.1. Let Ω be a bounded open subset of Rn. We denote by Cm,α(Ω)

the subspace of Cm(Ω) whose functions have mth order derivatives that are

Holder continuous with exponent α ∈ (0, 1].

6

Notation

Definition 0.2. By definition, a function f belongs to C∞(Ω) if f ∈ C∞(Ω)

and for all x ∈ ∂Ω and for all α ∈ Nn there exists the limit

limy→x

y∈ΩDαf(y).

(By definition Dαf(x) = limy→x

y∈ΩDαf(y).)

Definition 0.3. Let p ∈ [1,+∞]. Let λ > 0, and m, k be two natural numbers

such that m < λ < m + k. Then f ∈ Hλp (Nikol’skii space) if and only if

f ∈ Lp and

sup|α|=m, |h|=0

|h|m−λ∥khD

αf∥Lp < +∞.

This space does not depend on the choice of the integers k,m satisfying

the inequality m < λ < m+ k. We recall that Hλp is also known as the Besov

space Bλp,∞.

Definition 0.4. Let V,Ω be open subsets of Rn. We write

V ⊂⊂ Ω

if V ⊂ V ⊂ Ω and V is compact, and say that V is compactly embedded in Ω.

Definition 0.5. Let X and Y be normed space. By L(X ,Y) we denote the

normed space of the continuous linear maps of X to Y equipped with the topol-

ogy of uniform convergence on the unit sphere of X .

7

Chapter 1

Morrey and Sobolev Morrey

spaces

1.1 General Morrey spaces

Definition 1.1. Let Ω be a Lebesgue measurable subset of Rn. Let 0 < p ≤

+∞ and let w be a measurable function from ]0,+∞[ to ]0,+∞[. Denote by

Mw(·)p (Ω) the space of all real-valued measurable functions on Ω for which

∥f∥Mw(·)p (Ω)

= supx∈Ω

∥w(ρ)∥f∥Lp(B(x,ρ)∩Ω)∥L∞(0,∞) <∞.

Definition 1.2. Let 0 < p ≤ +∞. Denote by Λp,∞ the set of all measurable

functions w from ]0,+∞[ to ]0,+∞[ which are not equivalent to 0 such that

∥w(ρ)∥L∞(1,∞) <∞, ∥w(ρ)ρnp ∥L∞(0,1) <∞.

In [15], [18] it is proved that, if w is a non-negative measurable function

from ]0,+∞[ to ]0,+∞[ which are not equivalent to 0, then the spaceMw(·)p (Ω)

is non-trivial, i.e. consists not only of functions f equivalent to 0 on Ω if, and

only if, w ∈ Λp,∞.

Definition 1.3. If wλ(ρ) =

ρ−λ, ρ ∈]0, 1],

1, ρ ≥ 1,, then we set

Mλp (Ω) ≡ Mwλ

p (Ω)

and the condition wλ ∈ Λp,∞ means that 0 ≤ λ ≤ np.

9

1. Morrey and Sobolev Morrey spaces

Note that

Lemma 1.4.

∥f∥Mλp (Ω) = max

supx∈Ω

sup0<ρ<1

ρ−λ∥f∥Lp(B(x,ρ)∩Ω), ∥f∥Lp(Ω)

. (1.1)

We find convenient to set

|f |ρ,w,p,Ω ≡ supx∈Ω

∥∥w(r)∥f∥Lp(B(x,r)∩Ω)

∥∥L∞(0,ρ)

∀ρ ∈]0,+∞[,

and

|f |ρ,λ,p,Ω ≡ |f |ρ,wλ,p,Ω

for all measurable functions f from Ω to ]0,+∞[ and for all functions w from

]0,+∞[ to ]0,+∞[. Clearly, |f |ρ,w,p,Ω ∈ [0,+∞].

Definition 1.5. Let Ω be an open subset of Rn. Let p ∈ [1,+∞].

(i) Let w be a function from ]0,+∞[ to ]0,+∞[. Then we define as gener-

alized little Morrey space with weight w and exponent p the subspace

Mw,0p (Ω) ≡

f ∈ Mw

p (Ω) : limρ→0

|f |ρ,w,p,Ω = 0

of Mw

p (Ω).

(ii) Let λ ∈ [0,+∞[. Then, in particular, the little Morrey space with expo-

nents λ, p is the subspace

Mλ,0p (Ω) ≡

f ∈Mλ

p (Ω) : limρ→0

|f |ρ,λ,p,Ω = 0

of Mλ

p (Ω).

Example 1.6. Let 0 < λ < n/p. Then

1) The function |x|α ∈Mλp (B(0, 1)) if and only if α ≥ λ− n

p.

2) The function |x|α ∈Mλ,0p (B(0, 1)) if and only if α > λ− n

p.

Lemma 1.7. Let Ω be an open subset of Rn. Let p ∈ [1,+∞]. Let w ∈ Λp,∞.

Then Mw,0p (Ω) is a closed proper subspace of Mw

p (Ω).

Proof. Let f ∈ Mwp (Ω). Let fjj∈N be a sequence inMw,0

p (Ω) which converges

to f in Mwp (Ω). We want to prove that f ∈ Mw,0

p (Ω).

10


Let ε > 0. Since fj → f as j → ∞, there exists N1 ∈ N such that

|f − fk|+∞,w,p,Ω <ε

2∀ k ≥ N1.

By definition 1.5, there exists δ > 0 such that if ρ ∈]0, δ[, then

|fN1 |ρ,w,p,Ω <ε

2.

Since f = f − fN1 + fN1 , we have

|f |ρ,w,p,Ω ≤ |f − fN1 |+∞,w,p,Ω + |fN1 |ρ,w,p,Ω <ε

2+ε

2= ε,

for all ρ ∈]0, δ[.

Lemma 1.8. Let Ω be a bounded open subset of Rn. Let p ∈ [1,+∞]. Then

Mr−λ

p (Ω) =Mλp (Ω). Moreover, the quasi-norm

∥f∥Mρ−λ

p (Ω)= sup

x∈Ωρ>0

ρ−λ∥f∥Lp(B(x,ρ)∩Ω) ∀f ∈ Mρ−λ

p (Ω)

is equivalent to the quasi-norm

∥f∥Mλp (Ω) = sup

x∈Ω∥wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω)∥L∞(0,∞) =

= supx∈Ωρ>0

wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ∀f ∈Mλp (Ω).

Proof. A simple calculation shows that

∥f∥Mρ−λ

p (Ω)≤ ∥f∥Mλ

p (Ω).

Indeed, ρ−λ ≤ wλ(ρ) for all ρ ∈]0,+∞[ and thus

∥f∥Mρ−λ

p (Ω)= sup

(x,ρ)∈Ω×]0,+∞[

ρ−λ∥f∥Lp(B(x,ρ)∩Ω) ≤

≤ sup(x,ρ)∈Ω×]0,+∞[

wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) = ∥f∥Mλp (Ω).

Conversely,


(x,ρ)∈Ω×]0,+∞[

wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ≤

≤ sup

sup

(x,ρ)∈Ω×]0,1]ρ−λ∥f∥Lp(B(x,ρ)∩Ω), sup

(x,ρ)∈Ω×]1,+∞[

1 · ∥f∥Lp(B(x,ρ)∩Ω)

=

11


= sup

sup

(x,ρ)∈Ω×]0,1]ρ−λ∥f∥Lp(B(x,ρ)∩Ω), sup

(x,ρ)∈Ω×]1,+∞[

∥f∥Lp(B(x,ρ)∩Ω)

.

Now we estimate

sup(x,ρ)∈Ω×]1,+∞[

∥f∥Lp(B(x,ρ)∩Ω).

If 1 < ρ < diamΩ, then

∥f∥Lp(B(x,ρ)∩Ω) ≤ (diamΩ)λ(diamΩ)−λ∥f∥Lp(B(x,ρ)∩Ω) ≤

≤ (diamΩ)λ sup(x,ρ)∈Ω×]1, diamΩ[


≤ (diamΩ)λ sup(x,ρ)∈Ω×]0,+∞[

ρ−λ∥f∥Lp(B(x,ρ)∩Ω) = (diamΩ)λ∥f∥Mρ−λ

p (Ω).

If ρ ≥ sup1, diamΩ, then

∥f∥Lp(B(x,ρ)∩Ω) = ∥f∥Lp(Ω) ≤

≤ (diamΩ)λ(diamΩ)−λ∥f∥Lp(B(x, diamΩ)∩Ω) ≤

≤ (diamΩ)λ sup(x,ρ)∈Ω×]0,+∞[

ρ−λ∥f∥Lp(B(x,ρ)∩Ω) = (diamΩ)λ∥f∥Mρ−λ

p (Ω).

Therefore,

∥f∥Mλp (Ω) ≤ max1, (diamΩ)λ∥f∥

Mρ−λp (Ω)

,

and proof is complete.

Lemma 1.9. Let Ω be an open subset of Rn. Let 1 ≤ p ≤ +∞, 0 < λ < n/p.

Then Mλp (Ω) ⊆ Lp(Ω).

Proof. Let f ∈Mλp (Ω). We note that

wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ≤ ∥f∥Mλp (Ω) <∞ for all (x, ρ) ∈ Ω×]0,+∞[.

Since wλ(ρ) = 1 for ρ ≥ 1, we obtain

∥f∥Lp(B(x,ρ)∩Ω) ≤ ∥f∥Mλp (Ω) <∞ for all (x, ρ) ∈ Ω× [1,+∞[.

By taking supremum in ρ ≥ 1 we get

∥f∥Lp(Ω) ≤ ∥f∥Mλp (Ω).

12


Lemma 1.10. Let Ω be an open subset of Rn. Let 0 < p < ∞, 0 < λ < np.

Then

Mλp (Ω) * Lloc

q (Ω)

for any q > p.

Proof. Without loss of generality, we can assume that 0 ∈ Ω and B(0, 1) ⊂ Ω.

Let

f(x) =

(2kkn−1)1p , x ∈ B

(0, 1

k

)\B

(0, 1

k− 2−kk−pλ−1

), k ∈ N,

0, otherwise.

Note that∣∣∣∣B(0, 1k)\B

(0,

1

k− 2−kk−pλ−1

)∣∣∣∣ = vn

((1

k

)n

−(1

k− 2−kk−λp−1

)n)≤

≤ n vn

(1

k

)n−1

2−kk−λp−1 = σn2−kk−n−λp−2, (1.2)

where vn, σn respectively, is the volume, the surface area respectively, of the

unit ball in Rn. Similarly, since 1k− 2−kk−λp−1 > 1

2k,∣∣∣∣B(0, 1k

)\B

(0,

1


)∣∣∣∣ ≥ 21−nσn2−kk−n−λp−2. (1.3)

By using inequality (1.2) and the inequality

∑k≥a

1

kα+1≤(1 +

1

α

)1

aα, where α > 0, a ≥ 1,

we get that for any 0 < r ≤ 1 and x ∈ Rn

∥f∥pLp(B(x,r)) ≤ ∥f∥pLp(B(0,r)) =

=∑

1k−2−kk−λp−1≤r

2kkn−1∣∣∣∣B(0, 1k

)\B

(0,

1


)∣∣∣∣ ≤≤ σn

∑k≥ 1

2r

1

kλp+1≤ σn

(1 +

1

λp

)2λprλp.

If r ≥ 1 and x ∈ Rn, then

∥f∥pLp(B(x,r)) ≤ ∥f∥pLp(B(0,r)) = ∥f∥pLp(B(0,1)) = σn

(1 +

1

λp

)2λp.

Therefore, f ∈Mλp (Ω).

13


On the other hand, by (1.3) for any q > p

∥f∥qLq(B(0,r)) ≥∑1k≤r

(2kkn−1)qp

∣∣∣∣B(0, 1k)\B

(0,

1


)∣∣∣∣ ≥≥ 21−nσn

∑k≥ 1

r

2k(qp−1)k(n−1)

qp−n−λp−2 = ∞.

Hence, f /∈ Llocq (Ω).

Corollary 1.11. Let Ω be an open subset of Rn. Let 0 < p < ∞, 0 < λ < np.

Then

Hλp (Ω) ⊂Mλ

p (Ω)

and this inclusion is strict.

Proof. The above inclusion was proved in [29] for n = 1 and in [49] for n > 1.

The strictness of the inclusion follows since by the embedding theorem [48]

Hλp (Ω) ⊂ Lloc

q (Ω)

with q = npn−λp > p. Hence, the function f constructed in the proof of the

previous Lemma belongs to Mλp (Ω) but does not belong to Hλ

p (Ω).

Lemma 1.12. Let Ω be a Lebesgue measurable subset of Rn, mn(Ω) < ∞.

Let p ∈ [1,+∞[. Then the following statements hold.

(i) If λ ≤ np, then L∞(Ω) ⊆Mλ

p (Ω).

(ii) If λ < np, then L∞(Ω) ⊆Mλ,0

p (Ω).

Proof. (i) Let f ∈ L∞(Ω). Then we note that

∥f∥Lp(B(x,r)∩Ω) ≤ (mn(B(x, r) ∩ Ω))1p∥f∥L∞(Ω)

and


x∈Ωsupρ>0


≤ max

supx∈Ω

sup0<ρ≤1

ρ−λ(vnρn)

1p∥f∥L∞(Ω),

supx∈Ω

supρ>1

(mn(Ω))1p∥f∥L∞(Ω)

=

= max

v

1pn , (mn(Ω))

1p

∥f∥L∞(Ω).

14


(ii) Let f ∈ L∞(Ω). Then for all ρ ∈]0, 1] we consider the norm

|f |ρ,wλ,p,Ω = sup(x,r)∈Ω×]0,ρ[

wλ(r)∥f∥Lp(B(x,r)∩Ω) ≤

≤ sup(x,r)∈Ω×]0,ρ[

wλ(r)(mn(B(x, r) ∩ Ω))1p∥f∥L∞(Ω) ≤

≤ sup(x,r)∈Ω×]0,ρ[

r−λ(mn(B(x, r)))1p∥f∥L∞(Ω) ≤

≤ sup(x,r)∈Ω×]0,ρ[

rnp−λv

1pn ∥f∥L∞(Ω) → 0 as ρ→ 0.

Hence, we obtain that f ∈Mλ,0p (Ω).

Corollary 1.13. Let Ω be a Lebesgue measurable subset of Rn. Let p ∈

[1,+∞[, λ ∈ [0, n/p]. If f ∈ L∞(Ω) and supp f is compact, then f ∈Mλp (Ω).

Proof. Let f ∈ L∞(Ω). Then


x∈Ωsupρ>0


≤ supx∈Ω

supρ>0

wλ(ρ)(mn(B(x, r) ∩ supp f))1p∥f∥L∞(Ω) ≤

≤ max

supx∈Ω

sup0<ρ≤1

ρ−λ(vnρn)

1p∥f∥L∞(Ω),

supx∈Ω

supρ>1

(mn(supp f))1p∥f∥L∞(Ω)

=

= max

v

1pn , (mn(supp f))

1p

∥f∥L∞(Ω).

Corollary 1.14. Let Ω be an open subset of Rn. Let p ∈ [1,+∞[, λ ∈ [0, n/p].

Then C∞c (Ω) ⊂Mλp (Ω).

Next we state a known result for Morrey spaces. For the sake of complete-

ness we also give proofs.

Theorem 1.15. Let Ω be a bounded open subset of Rn. If 1 ≤ p < +∞ then

the following statements hold.

(i) M0p (Ω) = Lp(Ω);

(ii) If λ = np, then Mλ

p (Ω) = L∞(Ω) both algebraically and topologically;

15


(iii) If np< λ ≤ +∞, 1 ≤ p < +∞, then Mλ

p (Ω) = 0.

(iv) Let 0 < p ≤ q ≤ +∞ and 0 ≤ λ ≤ np, 0 ≤ ν ≤ n

q. If n

q− ν ≤ n

p− λ, then

M νq (Ω) →Mλ

p (Ω).

Proof. (i) Let us take f ∈M0p (Ω) and consider its norm in this space

∥f∥M0p (Ω) = sup

x∈Ωρ>0

w0(ρ)∥f∥Lp(B(x,ρ)∩Ω) = supx∈Ωρ>0

∥f∥Lp(B(x,ρ)∩Ω) = ∥f∥Lp(Ω).

(ii) If λ = np, then by the Lebesgue Theorem

∥f∥M

npp (Ω)

= supx∈Ωρ>0

ρ−np ∥f∥Lp(B(x,ρ)∩Ω) = υ

1pn sup

x∈Ωρ>0

∫

B(x,ρ)∩Ω|f(y)|pdy

υnρn

1p

≥

≥ υ1pn ess sup |f(x)| = υ

1pn ∥f∥L∞(Ω),

where υn is the volume of the unit ball in the space Rn.

Suppose now that f ∈ Mnpp (Ω) and f /∈ L∞(Ω). Since f /∈ L∞(Ω), i.e.

∥f∥L∞(Ω) = ∞, then for every K > 0 the set

S(f,K) = x ∈ Ω: |f(x)| > K

has positive measure.

Denote by A the set of all points x ∈ Ω for which

|f(x)|p = limρ→0+

1

mn(B(x, ρ))

∫B(x,ρ)

|f(y)|pdy.

Thus, A ∩ S(f,K) has a positive measure.

If x ∈ A ∩ S(f,K), then

limρ→0+

1

mn(B(x, ρ))

∫B(x,ρ)∩Ω

|f(y)|pdy = |f(x)|p > Kp.

From this fact it easy to see that for every K > 0 there exists ρ such

that1

mn(B(x, ρ))

∫B(x,ρ)∩Ω

|f(y)|pdy > K,

16


and, thus,

K1p < v

− 1p

n ρ−np

∫B(x,ρ)∩Ω

|f(y)|pdy

1p

≤ v− 1

pn sup

x∈Ωρ>0

ρ−np ∥f∥Lp(B(x,ρ)∩Ω).

So we have ∥f∥M

npp (Ω)

= ∞. This contradicts the assumptions that

f ∈ Mnpp (Ω).

Now let f ∈ L∞(Ω). For every ordered pair (x, ρ) ∈ Ω×]0,+∞[ we have

ρ−np

∫B(x,ρ)∩Ω

|f(y)|pdy

1p

≤ ρ−np

∫B(x,ρ)∩Ω

dy

1p

∥f∥L∞(Ω) ≤

≤ ρ−np υ

1pn ρ

np ∥f∥L∞(Ω) = υ

1pn ∥f∥L∞(Ω) ∀ρ ∈]0,+∞[.

Thus,

∥f∥M

npp (Ω)

≤ υ1pn ∥f∥L∞(Ω).

From this inequality follows the continuity of the identity operator

I : L∞(Ω) →Mnpp (Ω).

(iii) Let f ∈Mλp (Ω), then exploiting the Lebesgue Theorem we obtain


x∈Ωρ>0

ρ−λ∥f∥Lp(B(x,ρ)∩Ω) =

= supx∈Ωρ>0

ρ−λ+np

∫

B(x,ρ)∩Ω|f |pdx

ρn

1p

< +∞ ⇔ f(x) = 0 a.e.

17


(iv) Let f ∈M νq (Ω), then

∥f∥Mρ−λ

p (Ω)= sup

x∈Ωρ>0


≤ supx∈Ωρ>0

ρ−λ[mn(B(x, ρ) ∩ Ω)]1p− 1

q ∥f∥Lq(B(x,ρ)∩Ω) ≤

≤ max

sup

(x,ρ)∈Ω×]0,1]ρ−λ[mn(B(x, ρ))]

1p− 1

q ∥f∥Lq(B(x,ρ)∩Ω),

sup(x,ρ)∈Ω×[1,+∞[

ρ−λ[mn(Ω)]1p− 1

q ∥f∥Lq(B(x,ρ)∩Ω)

≤

≤ max

sup

(x,ρ)∈Ω×]0,1]υ

1p− 1

qn ρ−λ+

np−n

q ∥f∥Lq(B(x,ρ)∩Ω),

[mn(Ω)]1p− 1

q sup(x,ρ)∈Ω×[1,+∞[

wν(ρ)∥f∥Lq(B(x,ρ)∩Ω)

≤

≤ max

υ

1p− 1

qn , [mn(Ω)]

1p− 1

q

∥f∥Mν

q (Ω),

where υn is the volume of the unit ball in the space Rn.

Theorem 1.16. Let Ω be an open subset of Rn. Let p1, p2 ∈ [1,+∞] be such

that 1p1

+ 1p2

= 1p. Let λ1, λ2 ∈ [0,+∞[, λ = λ1 + λ2. Then the pointwise

multiplication is bilinear and continuous from Mλ1p1(Ω) × Mλ2

p2(Ω) to Mλ

p (Ω)

and maps Mλ1,0p1

(Ω)×Mλ2p2(Ω) to Mλ,0

p (Ω) and Mλ1p1(Ω)×Mλ2,0

p2(Ω) to Mλ,0

p (Ω)

Remark 1.17. This statement proves the Holder inequality for Morrey space

Mλp (Ω):

∥fg∥Mλp (Ω) ≤ ∥f∥

Mλ1p1

(Ω)∥g∥

Mλ2p2

(Ω)∀ (f, g) ∈Mλ1

p1(Ω)×Mλ2

p2(Ω).

Proof. Note that

wλ(ρ) =

ρ−λ, ρ ∈]0, 1],

1, ρ ≥ 1,=

ρ−λ1−λ2 , ρ ∈]0, 1],

1, ρ ≥ 1,= wλ1(ρ)wλ2(ρ).

Then, by Holder inequality, we have

|fg|ρ,λ,p,Ω = sup(x,r)∈Ω×]0,ρ[

wλ(r)∥fg∥Lp(B(x,r)∩Ω) ≤

≤ sup(x,r)∈Ω×]0,ρ[

wλ(r)∥f∥Lp1 (B(x,r)∩Ω)∥g∥Lp2 (B(x,r)∩Ω) ≤

18


≤ sup(x,r)∈Ω×]0,ρ[

wλ1(r)∥f∥Lp1 (B(x,r)∩Ω) sup(x,r)∈Ω×]0,ρ[

wλ2(r)∥g∥Lp2 (B(x,r)∩Ω) =

= |f |ρ,λ1,p1,Ω|g|ρ,λ2,p2,Ω for all ρ ∈]0,+∞].

Therefore, by taking ρ = +∞, we deduce that fg ∈ Mλp (Ω) when

(f, g) ∈ Mλ1p1(Ω)×Mλ2

p2(Ω).

By letting ρ→ 0, we deduce that fg ∈Mλ,0p (Ω) when (f, g) ∈ Mλ1,0

p1(Ω)×

Mλ2p2(Ω).

The case when f ∈Mλ1p1(Ω) and g ∈Mλ2,0

p2(Ω) can be analyzed in the same

way.

Theorem 1.18. Let Ω be an open subset of Rn. Let p ∈ [1,+∞]. Let

λ ∈ [0, n/p]. Then the pointwise multiplication is bilinear and continuous

from Mλp (Ω)× L∞(Ω) to M

λp (Ω) and maps Mλ,0

p (Ω)× L∞(Ω) to Mλ,0p (Ω).

Proof. We show that if f ∈Mλp (Ω), g ∈ L∞(Ω), then

|fg|ρ,λ,p,Ω = sup(x,r)∈Ω×]0,ρ[

wλ(r)∥fg∥Lp(B(x,r)∩Ω) ≤

≤ ∥g∥L∞(Ω) sup(x,r)∈Ω×]0,ρ[

wλ(ρ)∥f∥Lp(B(x,r)∩Ω) =

= ∥g∥L∞(Ω)|f |ρ,λ,p,Ω for all ρ ∈]0,+∞].

Hence, by taking ρ = +∞, we deduce that fg ∈ Mλp (Ω) when

(f, g) ∈ Mλp (Ω)× L∞(Ω).

By letting ρ→ 0, we deduce that fg ∈Mλ,0p (Ω) when (f, g) ∈ Mλ,0

p (Ω)×

L∞(Ω).

Lemma 1.19. Let A ⊂ Rm be a measurable set. Let Ω be an open subset of Rn.

Let p ∈ [1,+∞]. Let λ ∈ [0, n/p]. Suppose that f is a measurable from A× Ω

to R. Let f(·, y) ∈Mλp (Ω) for almost all y ∈ A and

∫A

∥f(·, y)∥Mλp (Ω) dy < +∞.

Then for almost all x ∈ Ω the integral∫A

f(x, y)dy makes sense and Minkowski’s

inequality for Morrey spaces∥∥∥∥∥∥∫A

f(·, y)dy

∥∥∥∥∥∥Mλ

p (Ω)

≤∫A

∥f(·, y)∥Mλp (Ω) dy

holds.

19


Proof. By Minkowski’s inequality for the Lebesgue spaces and by the imbed-

ding of Mλp (Ω) into Lp(Ω) we know that for almost all x ∈ Ω the integral∫

A

f(x, y)dy makes sense and defines almost everywhere a function of Lp(Ω).

Then by applying the Minkowski’s inequality for the Lebesgue spaces in

(B(x, ρ) ∩ Ω)× A for all ρ ∈]0,+∞[, we obtain the following inequality∥∥∥∥∥∥∫A

f(·, y)dy

∥∥∥∥∥∥Mλ

p (Ω)

= sup(x,ρ)∈Ω×]0,+∞[

wλ(ρ)

∥∥∥∥∥∥∫A

f(·, y)dy

∥∥∥∥∥∥Lp(B(x,ρ)∩Ω)

≤

≤ sup(x,ρ)∈Ω×]0,+∞[

wλ(ρ)

∫A

∥f(·, y)∥Lp(B(x,ρ)∩Ω) dy =

=

∫A

sup(x,ρ)∈Ω×]0,+∞[

wλ(ρ) ∥f(·, y)∥Lp(B(x,ρ)∩Ω) dy =

∫A

∥f(·, y)∥Mλp (Ω) dy.

1.2 Approximation by C∞ functions in Morrey

spaces

Definition 1.20. If ϕ ∈ L1(Rn) and t ∈]0,+∞[, we denote by ϕt(·) the func-

tion from Rn to R defined by

ϕt(x) ≡ t−nϕ(x/t) ∀ x ∈ Rn.

By the formula of change of variables in integrals, we conclude that∫Rn

ϕt(x)dx =

∫Rn

ϕ(x)dx ∀ t ∈]0,+∞[,

whenever ϕ ∈ L1(Rn).

Lemma 1.21. Let p ∈ [1,+∞[, 0 ≤ λ < npand f ∈Mλ,0

p (Rn). Then

limk→∞

fχB(0,k) = f in Mλp (Rn).

Proof. Consider for 0 < ρ ≤ 1 the norm

∥fχB(0,k) − f∥Mλp (Rn) = sup

x∈Rn

∥wλ(r)∥fχB(0,k) − f∥Lp(B(x,r))∥L∞(0,∞) =

= supx∈Rn

∥wλ(r)∥f∥Lp(B(x,r)\B(0,k))∥L∞(0,∞) ≤

20

1.2 Approximation by C∞ functions in Morrey spaces

≤ supx∈Rn

∥wλ(r)∥f∥Lp(B(x,r))∥L∞(0,ρ) +maxρ−λ, 1∥f∥Lp(Rn\B(0,k)) =

= |f |ρ,λ,p,Rn + ρ−λ∥f∥Lp(Rn\B(0,k)).

Let k → ∞, then since f ∈ Lp(Rn) (by Lemma 1.9) and p <∞ we have

limk→∞

∥f∥Lp(Rn\B(0,k)) = 0.

Therefore, for all 0 < ρ ≤ 1

limk→∞

∥fχB(0,k) − f∥Mλp (Rn) ≤ |f |ρ,λ,p,Rn .

By passing to the limit as ρ→ 0, since f ∈Mλ,0p (Rn), we have

limk→∞

∥fχB(0,k) − f∥Mλp (Rn) = 0,

or equivalently

limk→∞

∥fχB(0,k) − f∥Mλp (Rn) = 0.

Then we have the following result of approximation by convolution.

Theorem 1.22. Let ϕ ∈ C∞c (Rn),∫Rn

ϕ(x)dx = 1. Then the following state-

ments hold.

(i) Let p ∈ [1,+∞], λ ∈[0, n

p

]. If f ∈Mλ

p (Rn) and ε > 0, then the function

f ∗ ϕε from Rn to R defined by

f ∗ ϕε ≡∫Rn

f(x− y)ϕε(y)dy ∀ x ∈ Rn

belongs to Mλp (Rn) ∩ C∞(Rn) and

∥f ∗ ϕε∥Mλp (Rn) ≤ ∥ϕ∥L1(Rn)∥f∥Mλ

p (Rn) ∀ f ∈Mλp (Rn).

(ii) Let p ∈ [1,+∞], λ ∈[0, n

p

]. If f ∈ Mλ,0

p (Rn) and ε > 0, then f ∗ ϕε

belongs to Mλ,0p (Rn) ∩ C∞(Rn).

(iii) Let p ∈ [1,+∞[. If f ∈ Mλ,0p (Rn), then f ∗ ϕε belongs to Mλ,0

p (Rn) ∩

C∞(Rn) ∩ C0ub(Rn) for all ε ∈]0,+∞[ and

limε→0

f ∗ ϕε = f in Mλp (Rn). (1.4)

21


(iv) Let p ∈ [1,+∞[. Then

clMλp (Rn)C

∞c (Rn) =Mλ,0

p (Rn). (1.5)

Proof. (i) Let f ∈Mλp (Rn) and ε > 0, then

wλ(ρ)∥f ∗ ϕε∥Lp(B(x,ρ)) ≤ wλ(ρ)

∥∥∥∥∥∥∫Rn

f(ξ − y)ϕε(y)dy

∥∥∥∥∥∥Lp,ξ(B(x,ρ))

≤

≤ wλ(ρ)

∫B(0,ε)

∥f(ξ − y)∥Lp,ξ(B(x,ρ))|ϕε(y)|dy =

= wλ(ρ)

∫Rn

|ϕε(y)|dy

supy∈B(0,ε)

∥f(ξ − y)∥Lp,ξ(B(x,ρ)) =

= wλ(ρ)

∫Rn

|ϕ(y)|dy

supy∈B(0,ε)

∥f(z)∥Lp(B(x−y,ρ)) =

= wλ(ρ)

∫Rn

|ϕ(y)|dy

supz∈B(x,ε)

∥f∥Lp(B(z,ρ)) ≤

≤ wλ(ρ)

∫Rn

|ϕ(y)|dy

supz∈Rn

∥f∥Lp(B(z,ρ)) for all ρ ∈]0,+∞[.

Thus, we have

∥f ∗ ϕε∥Mλp (Rn) ≤

∫Rn

|ϕ|dx

∥f∥Mλp (Rn) ∀ f ∈Mλ

p (Rn).

(ii) Let f ∈ Mλ,0p (Rn) and ε > 0. In the proof of statement (i) we have

proved that

wλ(r)∥f ∗ ϕε∥Lp(B(x,r)) ≤

wλ(r)

∫Rn

|ϕ(x)|dx

supx∈Rn

∥f∥Lp(B(x,r)) for all r ∈]0,+∞[.

Moreover,

wλ(r)∥f∥Lp(B(x,r)) ≤ |f |ρ,λ,p,Rn ∀x ∈ Rn, r ∈]0, ρ[,

and thus, by taking the supremum on x ∈ Rn, we have

wλ(r) supx∈Rn

∥f∥Lp(B(x,r)) ≤ |f |ρ,λ,p,Rn ∀ r ∈]0, ρ[.

22


Hence,

supr∈]0,ρ[


∫Rn

|ϕ(x)|dx

|f |ρ,w,p,Rn ,

and

limρ→0

|f ∗ ϕε|ρ,λ,p,Rn = 0 ∀ f ∈Mλ,0p (Rn).

(iii) Let η > 0. Since f ∈Mλ,0p , there exists ρη > 0 such that

|f |ρ,λ,p,Rn ≤ η

1 +

∫Rn

|ϕ(y)|dy

−1 ∀ ρ ∈]0, ρη].

Then we have

sup(x,r)∈Rn×]0,ρη [

wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤

≤ sup(x,r)∈Rn×]0,ρη [

wλ(r)∥f∥Lp(B(x,r)) + sup(x,r)∈Rn×]0,ρη [


≤ |f |ρη ,λ,p,Rn +

∫Rn

|ϕ(y)|dy

|f |ρη ,λ,p,Rn =

= |f |ρη ,λ,p,Rn

1 +

∫Rn

|ϕ(y)|dy

≤ η, (1.6)

for all ε ∈]0,+∞[.

Further, we have

sup(x,r)∈Rn×[ρη ,+∞[

wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤

≤

(sup

r∈[ρη ,+∞[

wλ(r)

)∥f − f ∗ ϕε∥Lp(Rn) for all ε > 0.

Since f ∈ Lp(Rn) (by Lemma 1.9) and p ∈ [1,+∞[ and∫Rn

|ϕ(y)|dy = 1,

standard properties of approximate identities of convolution imply that

limε→0

∥f − f ∗ ϕε∥Lp(Rn) = 0.

Thus, there is exists εη > 0 such that(sup

r∈[ρη ,+∞[

wλ(r)

)∥f − f ∗ ϕε∥Lp(Rn) ≤ η ∀ ε ∈]0, εη].

23


So, we obtain

sup(x,r)∈Rn×[ρη ,+∞[

wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤ η ∀ ε ∈]0, εη]. (1.7)

By combining inequalities (1.6) and (1.7), we have

sup(x,r)∈Rn×[0,+∞[

wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤ η ∀ ε ∈]0, εη].

Hence,

limε→0

f ∗ ϕε = f in Mλp (Rn).

(iv) Clearly, C∞c (Rn) is contained in Mλ,0p (Rn). Since Mλ,0

p (Rn) is closed in

Mλp (Rn) (by Lemma 1.7), we obtain

clMλp (Rn)C

∞c (Rn) ⊂ clMλ

p (Rn)Mλ,0p (Rn) =Mλ,0

p (Rn).

Now let f ∈Mλ,0p (Rn). Then for all k ∈ N

fχB(0,k) ∗ ϕ 1k∈ C∞c (Rn).

Next, using (i) of Theorem 1.22, we obtain

∥fχB(0,k) ∗ ϕ 1k− f∥Mλ

p (Rn) ≤

≤ ∥(fχB(0,k) − f) ∗ ϕ 1k∥Mλ

p (Rn) + ∥f ∗ ϕ 1k− f∥Mλ

p (Rn) ≤

≤ ∥fχB(0,k) − f∥Mλp (Rn)∥ϕ∥L1(Rn) + ∥f ∗ ϕ 1

k− f∥Mλ

p (Rn).

By Lemma 1.21

limk→∞

∥fχB(0,k) − f∥Mλp (Rn) = 0,

and by (iii) of Theorem 1.22

limk→∞

∥f ∗ ϕ 1k− f∥Mλ

p (Rn) = 0.

So, we obtain that

limk→∞

∥fχB(0,k) ∗ ϕ 1k− f∥Mλ

p (Rn) = 0.

Thus, f ∈ clMλp (Rn)C

∞c (Rn) and equality (1.5) holds.

24


Remark 1.23. The limiting relation (1.4) does not hold for all f ∈ Mλp (Rn).

For example,

|x|λ−npχB(0,1) ∗ ϕε 9 |x|λ−

npχB(0,1) in Mλ

p (Rn) (1.8)

as ε→ 0.

Indeed, function |x|λ−npχB(0,1) ∗ ϕε belongs to C∞c (Rn). If

|x|λ−npχB(0,1) ∗ ϕε → |x|λ−

npχB(0,1) in Mλ

p (Rn)

then by (iv) of Theorem 1.22 |x|λ−npχB(0,1) ∈ clMλ

p (Rn)C∞c (Rn) = Mλ,0

p (Rn),

which is not true (see Example 1.6).

Lemma 1.24. Let Ω be an open subset of Rn. Let EΩ be the extension operator

from M(Ω) to M(Rn) defined by

EΩf ≡

f, in Ω,

0, in Rn \ Ω,

Let p ∈ [1,+∞], λ ∈ [0, n/p]. Then for all f ∈Mλp (Ω)

|EΩf |ρ,wλ,p,Rn ≤ 2λ|f |2ρ,wλ,p,Ω (1.9)

for all ρ ∈]0,+∞] and

∥EΩf∥Mλp (Rn) ≤ 2λ∥f∥Mλ

p (Ω). (1.10)

In particular, EΩ maps Mλp (Ω) to M

λp (Rn) and Mλ,0

p (Ω) to Mλ,0p (Rn).

Proof. Let ρ ∈]0,+∞]. Then we have

|EΩf |ρ,wλ,p,Rn = sup(x,r)∈Rn×]0,ρ[

wλ(r)∥EΩf∥Lp(B(x,r)) =

= sup0<r<ρ

supx∈Rn

wλ(r)∥f∥Lp(B(x,r)∩Ω) =

= sup0<r<ρ

supx∈Rn

B(x,r)∩Ω=∅

wλ(r)∥f∥Lp(B(x,r)∩Ω).

If (x, r) ∈ Rn×]0, ρ[ and B(x, r)∩Ω = ∅, then there exists ξ(x) ∈ B(x, r)∩

Ω. By the triangle inequality, we have

B(x, r) ∩ Ω ⊂ B(ξ(x), 2r) ∩ Ω,

25


hence,

|EΩf |ρ,wλ,p,Rn ≤ sup0<r<ρ

supx∈Rn

B(x,r)∩Ω=∅

wλ(r)∥f∥Lp(B(ξ(x),2r)∩Ω) ≤

≤ sup0<r<ρ

supη∈Ω

wλ(r)∥f∥Lp(B(η,2r)∩Ω).

We also note that

wλ(ρ) ≤ 2λwλ(2ρ) ∀ ρ ∈ [0,+∞[.

Therefore,

|EΩf |ρ,wλ,p,Rn ≤ 2λ supη∈Ω

sup0<r<ρ

wλ(2r)∥f∥Lp(B(η,2r)∩Ω) = 2λ|f |2ρ,wλ,p,Ω.

Inequality (1.10) follows by inequality (1.9).

Theorem 1.25. Let Ω be an open subset of Rn. Let p ∈ [1,+∞[. Then the

following statements hold.

(i) clMλp (Ω)

(Mλ,0

p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)

)=

= clMλp (Ω)

(Mλ,0

p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)

)=Mλ,0

p (Ω).

(ii) If mn(Ω) <∞ and if λ < np, then C0

ub(Ω) ⊆ L∞(Ω) ⊆Mλ,0p (Ω) and

clMλp (Ω)

(C∞(Ω) ∩ C0

ub(Ω))= clMλ

p (Ω)

(C∞(Ω) ∩ C0

ub(Ω))=Mλ,0

p (Ω).

(iii) If Ω is bounded and if λ < np, then

C∞(Ω) ⊆ C∞(Ω) ⊆ C0ub(Ω) ⊆ L∞(Ω) ⊆Mλ,0

p (Ω)

and

clMλp (Ω)C

∞(Ω) = clMλp (Ω)C

∞(Ω) =Mλ,0p (Ω).

Proof. (i) First we observe that the sets Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0

ub(Ω) and

Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0

ub(Ω) are contained in Mλ,0p (Ω).

By Lemma 1.7 Mλ,0p (Ω) is a closed subspace of Mλ

p (Ω).

Then the closure of the set Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0

ub(Ω) is contained in

Mλ,0p (Ω).

26


Similarly, the closure of the set Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0

ub(Ω) is contained

in Mλ,0p (Ω).

Let now f ∈ Mλ,0p (Ω). Then, by Lemma 1.27, there exists a bounded

linear extension operator EΩ :Mλ,0p (Ω) →Mλ,0

p (Rn) such that EΩf∣∣Ω= f

for all f ∈Mλ,0p (Ω).

Thus, we can approximate EΩf by smooth functions, which are, by def-

inition, functions from C∞(Ω).

Therefore, EΩf , as a limit, belongs to

clMλp (Ω)

(Mλ,0

p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)

)=

= clMλp (Ω)

(Mλ,0

p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)

).

(ii) Let f ∈ C0ub(Ω), then f is bounded and, thus, it is essentially bounded,

i.e. f ∈ L∞(Ω) and hence

C0ub(Ω) ⊆ L∞(Ω).

Then by Lemma 1.12 (ii) we have L∞(Ω) ⊂Mλ,0p (Ω). Hence,

Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0

ub(Ω) = C∞(Ω) ∩ C0ub(Ω)

and

Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0

ub(Ω) = C∞(Ω) ∩ C0ub(Ω).

Therefore, from (i) we obtain

clMλp (Ω)

(C∞(Ω) ∩ C0

ub(Ω))= clMλ

p (Ω)

(C∞(Ω) ∩ C0

ub(Ω))=Mλ,0

p (Ω).

(iii) Let f ∈ C∞(Ω). By definition there exist an open neighborhood U of Ω

and a function F ∈ C∞(U) such that F |Ω = f . Then f ∈ C∞(Ω).

Now let f ∈ C∞(Ω). Then f is continuous.

Since Ω is bounded and Ω is closed, then, by Cantor’s Theorem, f is

uniformly continuous. In this case f is also bounded. Thus, f ∈ C0ub(Ω).

Hence, we have

C∞(Ω) ∩ C0ub(Ω) = C∞(Ω) and C∞(Ω) ∩ C0

ub(Ω) = C∞(Ω).

27


From (ii) we get

clMλp (Ω)C

∞(Ω) = clMλp (Ω)C

∞(Ω) =Mλ,0p (Ω).

1.3 Preliminaries on integral operators

Lemma 1.26. Let f ∈ Lloc1 (Rn). If there exists R ∈]0,+∞[ such that

f∣∣Rn\Bn(0,R)

is essentially bounded, then for each ε > 0 there exists δ > 0

such that ∫E

|f | dmn ≤ ε ∀E ∈ Ln, mn(E) ≤ δ.

Proof. Since f∣∣Rn\Bn(0,R)

is essentially bounded, we have f ∈ L∞(Rn\Bn(0, R))

and f ∈ L1(Bn(0, R)).

We set

E1 = E ∩Bn(0, R), E2 = E \Bn(0, R), for all E ∈ Ln.

Now let ε > 0. Then, by absolute continuity of the Lebesgue integral, there

exists δ1(ε) > 0 such that ∫E1

|f | dmn ≤ ε

2

whenever mn(E1) ≤ δ1(ε).

Next we suppose that

δ2(ε) =ε

2∥f∥L∞(Rn\Bn(0,R)) + 1,

then for E2 satisfying mn(E2) ≤ δ2(ε) we obtain∫E2

|f | dmn ≤ ∥f∥L∞(Rn\Bn(0,R))mn(E2) < ∥f∥L∞(Rn\Bn(0,R))δ2(ε) =

= ∥f∥L∞(Rn\Bn(0,R))ε

2∥f∥L∞(Rn\Bn(0,R))

=ε

2.

Therefore,∫E

|f | dmn =

∫E1

|f | dmn +

∫E2

|f | dmn ≤ ε

2+ε

2= ε.

28


Lemma 1.27. Let n ∈ N \ 0. Let λ ∈]0, n[. For each ε > 0 there exists

δ > 0 such that ∫E

1

|ξ − η|λdη ≤ ε ∀E ∈ Ln, mn(E) ≤ δ.

for all ξ ∈ Rn.

Proof. Setting ξ − η = z, we have∫E

1

|ξ − η|λdη =

∫ξ−E

dz

|z|λ.

Function 1|x|λ ∈ L1(B(0, 1)) and 1

|x|λ ∈ L∞(Rn \Bn(0, 1)).

Then, by applying the previous lemma, we complete the proof.

Theorem 1.28. Let n ∈ N \ 0. Let α ∈]0, n[.Let Ω be an open subset of Rn

of finite measure. Let Ω1 be an open subset of Rn. Let

DΩ1×Ω ≡ (x, y) ∈ Ω1 × Ω : x = y .

Let k be a function from (Ω1×Ω)\DΩ1×Ω to R such that k(x, ·) is measurable

in Ω \ x for all x ∈ Ω1. Assume that there exists c ∈]0,+∞[ such that

|k(x, y)| ≤ c

|x− y|n−α∀ (x, y) ∈ (Ω1 × Ω) \DΩ1×Ω.

Assume that if x ∈ Ω1, there exists a subset Nx of measure 0 of Ω such that

x /∈ Ω \Nx and such that the function k(·, y) from Ω1 \ y to R is continuous

at x for all y ∈ Ω \Nx.

If f ∈ L∞(Ω), then the function H[f ] from Ω1 to R defined by

H[f ](x) ≡∫Ω

k(x, y)f(y)dy ∀ x ∈ Ω1

is continuous.

Proof. Let x ∈ Ω1. Clearly,

|k(x, y)f(y)| ≤ ∥f∥L∞(Ω)c

|x− y|n−α(1.11)

for almost all y ∈ Ω. Since c|x−y|n−α is integrable in y ∈ Ω, we deduce that

k(x, y)f(y) is integrable in y ∈ Ω.

29


Now let x ∈ Ω1. In order to prove the continuity of H[f ] at x, we want to

apply the Vitali Convergence Theorem

By inequality (1.11), we have∫E

|k(x, y)f(y)|dy ≤ c∥f∥L∞(Ω)

∫E

dy

|x− y|n−α≤

≤ c∥f∥L∞(Ω) supx∈Rn

∫E

dy

|x− y|n−α,

for all x ∈ Ω1. Now let ε > 0. Lemma 1.27 implies that there exists δ > 0

such that∫E

dy

|x− y|n−α≤ ε

1 + c∥f∥L∞(Ω)

if E ∈ Ln, mn(E) ≤ δ, x ∈ Rn.

Then we have∫E

|k(x, y)f(y)|dy ≤ ε if E ∈ Ln, mn(E) ≤ δ, x ∈ Ω1. (1.12)

Now let xjj∈N be a sequence in Ω1 \ x which converges to x in Ω1. Let

Nx be a subset of measure 0 of Ω as in assumptions. Then we have

limj→∞

k(xj, y) = k(x, y) ∀ y ∈ Ω \Nx, (1.13)

and, in particular, for almost all y ∈ Ω. Then (1.12) and (1.13) and the Vitali

Convergence Theorem imply that

limj→∞

H[f ](xj) = H[f ](x).

Hence, H[f ] is continuous at x.

Let f ∈ Lloc1 (Rn). Consider the Riesz potential

(Iαf)(x) =

∫Rn

f(y)

|x− y|n−αdy, 0 < α < n.

In [14], in particular, the following statement is proved generalizing the results

of [1], [17], [42], [47], [27] and [16].

Theorem 1.29. Let condition

1 < p ≤ ∞, 0 < q ≤ ∞ and n

(1

p− 1

q

)+

< α < n, (1.14)

30


or

p = 1, 0 < q <∞ and n

(1− 1

q

)+

< α < n, (1.15)

or

1 < p < q < +∞ and α = n

(1

p− 1

q

)(1.16)

be satisfied. Let also u ∈ Λp,∞, v ∈ Λq,∞ and

I(u, v) =

∥∥∥∥∥∥v(t)tnq∞∫t

sα−np−1

∥u∥L∞(s,∞)

ds

∥∥∥∥∥∥L∞(0,∞)

<∞. (1.17)

Then the operator Iα is bounded from Mu(·)p (Rn) to Mv(·)

q (Rn).

Moreover, if condition (1.15) is satisfied, then condition (1.17) is necessary

and sufficient for the boundedness of Iα from Mu(·)p (Rn) to Mv(·)

q (Rn).

Theorem 1.30. Let n ∈ N \ 0. Let 1 ≤ p ≤ q < +∞. Let 0 ≤ λ ≤ ν < nq.

Let

α ≡(ν − n

q

)−(λ− n

p

). (1.18)

Then the following statements hold.

(i) If λ < ν, then the operator Iα is bounded from Mr−λ

p (Rn) to Mr−ν

q (Rn).

(ii) If λ = ν and if 1 < p < q, then Iα is bounded from Mr−λ

p (Rn) to

Mr−ν

q (Rn).

(iii) If λ = ν and if 1 < p < q, then Iα is bounded from Mλp (Rn) to Mν

q (Rn).

Remark 1.31. If ν = λ = 0, then α = n(

1p− 1

q

), and this is the classical

Hardy-Littlewood-Sobolev theorem.

Proof. (i) We recall that

t+ =

t, if t ≥ 0,

0, if t < 0.

Since p ≤ q, we have n(

1p− 1

q

)+= n

(1p− 1

q

).

Assumptions λ < ν and (1.18) imply α =(ν − n

q

)−(λ− n

p

),

λ < ν,⇒

α = ν − λ+ np− n

q,

ν − λ > 0,⇒ n

(1

p− 1

q

)< α,

31


α =(ν − n

q

)−(λ− n

p

),

0 ≤ λ ≤ ν < nq,

⇒

α = ν − λ+ np− n

q,

ν − λ < nq,

⇒ α <n

p≤ n.

Therefore,

n

(1

p− 1

q

)< α < n,

and thus either condition (1.14) or (1.15) satisfied.

Now we want to prove that

I(r−λ, r−ν) = supt>0

r−ν(t)tnq

∞∫t

sα−np−1

r−λ(s)ds <∞.

Indeed,

t−ν+nq

∞∫t

sα+λ−np−1ds = t−ν+

nq

∞∫t

sν−nq−1ds =

=t−ν+

nq+ν−n

q

−ν + nq

=

(−ν + n

q

)−1<∞.

By Theorem 1.29, operator Iα is linear and continuous from Mr−λ

p (Rn)

to Mr−ν

q (Rn).

(ii) If λ = ν, then α = n(

1p− 1

q

)and, hence, condition (1.16) satisfied.

Now we prove that inequality (1.17) holds with u(t) = v(t) = t−λ. In-

deed,

t−λ+nq

∞∫t

sα+λ−np−1ds = t−λ+

nq

∞∫t

sλ−nq−1ds =

=t−λ+

nq+λ−n

q

−λ+ nq

=

(−λ+

n

q

)−1<∞.

Hence, Theorem 1.29 implies that Iα is continuous from Mr−λ

p (Rn) to

Mr−ν

q (Rn).

(iii) If λ = ν, then α = n(

1p− 1

q

)and, therefore, condition (1.16) satisfied.

We want to prove that

I(wλ, wλ) = supt>0

wλ(t)tnq

∞∫t

sα−np−1

wλ(s)ds <∞.

32


Let first t ∈]0, 1]. Then

t−λ+nq

∞∫t

sα+λ−np−1ds = t−λ+

nq

1∫t

sλ−nq−1ds+ t−λ+

nq

∞∫1

sλ−nq−1ds =

=t−λ+

nq − 1− t−λ+

nq

λ− nq

=

(−λ+

n

q

)−1<∞.

Now let t ≥ 1. Then wλ(t) = 1 and

tnq

∞∫t

sα−np−1ds = t

nq

∞∫t

s−nq−1ds =

q

n<∞.

Thus, (1.17) holds and Theorem 1.29 implies that Iα is continuous from

Mλp (Rn) to Mν

q (Rn).

Lemma 1.32. Let p ∈ [1,+∞[, α ∈]0, n[, λ ∈ [0, n/p]. Let q ∈ [1, p] be such

that (α + λ) > nq. Let

µwλ,q ≡ max

1,

1

(α + λ)− nq

. (1.19)

Then we have∫E∩Bn(x,1)

|f(y)|dy|x− y|n−α

≤

≤ mn(E)1q− 1

pµwλ,q(n+ 2− α)v1− 1

qn ∥f∥Mλ

p (Rn) ∀ f ∈Mλp (Rn), (1.20)

for all measurable subsets E of Rn of finite measure, and for all x ∈ Rn.

Proof. The arguments of this proof are in part based on a development of the

ideas of Campanato [19].

If f ∈ Mλp (Rn), then we know that f

∣∣Bn(x,r)

∈ Lp(Bn(x, r)) ⊆ L1(Bn(x, r))

for all x ∈ Rn and r ∈]0,+∞[. In particular, (χEf)∣∣Bn(x,r)

∈ L1(Bn(x, r)) for

all x ∈ Rn and r ∈]0,+∞[ and for all measurable subsets E of Rn.

Now we fix x ∈ Rn and a measurable subset E of Rn of finite measure.

The almost everywhere defined function from ]0,+∞[ to [0,+∞[ which takes

s ∈]0,+∞[ to∫

∂Bn(x,s)

χE|f |dσ is integrable in ]0, r[ for all r ∈]0,+∞[. Then

33


by the Fundamental Theorem of Calculus, the function AE,x from [0,+∞[ to

[0,+∞[ defined by

AE,x(ρ) ≡ρ∫

0

∫∂Bn(x,s)

χE|f |dσ ds ∀ ρ ∈ [0,+∞[,

is locally absolutely continuous and

A′E,x(ρ) ≡∫

∂Bn(x,ρ)

χE|f |dσ,

for almost all ρ ∈ [0,+∞[ (cf. e.g., Folland [25, 3.35]). By the Monotone

Convergence Theorem, we have

∫E∩Bn(x,1)


=

=

∫Bn(x,1)

χE(y)|f(y)|dy|x− y|n−α

= limε→0

∫Bn(x,1)\Bn(x,ε)


. (1.21)

Now let ε ∈]0, 1[. Then we have

∫Bn(x,1)\Bn(x,ε)


=

=

1∫ε

s−n+α

∫∂Bn(x,s)

χE|f |dσ ds =1∫

ε

s−n+αA′E,x(s)ds. (1.22)

Then by integrating by parts, we obtain

1∫ε

s−n+αA′E,x(s)ds =

=[s−n+αAE,x(s)

]1ε−

1∫ε

(−n+ α)s−n+α−1AE,x(s)ds, (1.23)

(cf. e.g., Folland [25, ex.35 p.108]). Then the Holder inequality and inequality

34


(1.19) imply that

|AE,x(ρ)| ≤ mn(E ∩ Bn(x, ρ))1− 1

p∥f∥Lp(E∩Bn(x,ρ)) =

= mn(E ∩ Bn(x, ρ))1q− 1

p mn(E ∩ Bn(x, ρ))1− 1

q ∥f∥Lp(E∩Bn(x,ρ)) ≤

≤ mn(E)1q− 1

p v1− 1

qn ρn−

nq ∥f∥Lp(Bn(x,ρ)) ≤

≤ mn(E)1q− 1

p v1− 1

qn ρn−αρ−n+αρn−

nqw−1λ (ρ)wλ(ρ)∥f∥Lp(Bn(x,ρ)) ≤

≤ ρn−αmn(E)1q− 1

p v1− 1

qn ρα−

nqw−1λ (ρ)∥f∥Mλ

p (Rn) ≤

≤ ρn−αmn(E)1q− 1

p v1− 1

qn ρ(α+λ)−n/q∥f∥Mλ

p (Rn) (1.24)

for all ρ ∈]0, 1[. Then by the second last line of inequality (1.24), we have∣∣∣∣∣∣1∫

ε

(−n+ α)s−n+α−1AE,x(s)ds

∣∣∣∣∣∣ ≤≤ mn(E)

1q− 1

p v1− 1

qn ∥f∥Mλ

p (Rn)

∣∣∣∣∣∣1∫

ε

(−n+ α)s−n+α−1sn−αs(α+λ)−nq ds

∣∣∣∣∣∣ ≤≤ mn(E)

1q− 1

p v1− 1

qn ∥f∥Mλ

p (Rn)(n− α)

∣∣∣∣∣∣1∫

ε

s(α+λ)−nq−1ds

∣∣∣∣∣∣ == mn(E)

1q− 1

p v1− 1

qn ∥f∥Mλ

p (Rn)(n− α)1

(α+ λ)− n/q(1− ε) =

= mn(E)1q− 1

p (n− α)v1− 1

qn µwλ,q∥f∥Mλ

p (Rn)(1− ε). (1.25)

Then by combining (1.22)–(1.25), we deduce that∫Bn(x,1)\Bn(x,ε)


≤

∣∣∣∣∣∣1∫

ε

s−n+αA′E,x(s)ds

∣∣∣∣∣∣ ≤

≤ |AE,x(1)|+ |ε−n+αAE,x(ε)|+

∣∣∣∣∣∣1∫

ε

(−n+ α)s−n+α−1AE,x(s)ds

∣∣∣∣∣∣ ≤≤ 1n−αmn(E)

1q− 1

p v1− 1

qn 1(α+λ)−n/q∥f∥Mλ

p (Rn)+

+ε−n+αεn−αmn(E)1q− 1

p v1− 1

qn ε(α+λ)−n/q∥f∥Mλ

p (Rn)+

+mn(E)1q− 1

p (n− α)v1− 1

qn µwλ,q∥f∥Mλ

p (Rn)(1− ε) ≤

≤ mn(E)1q− 1

pµwλ,qv1− 1

qn ∥f∥Mλ

p (Rn)[1 + 1 + (n− α)(1− ε)].

Then the limiting relation (1.21) immediately implies the validity of inequality

(1.20).

35


Corollary 1.33. Let p ∈ [1,+∞[, α ∈]0, n[. Let λ ∈ [0, n/p]. Let α + λ > np.

Let Ω be an open subset of Rn. Then the following statements hold.

If f ∈Mλp (Rn) and if

∫Ω

|f |dx < ∞, then the function from Rn to R which

takes x ∈ Rn to ∫Ω

f(y)dy

|x− y|n−α

is bounded, and satisfies the following inequality

supx∈Rn

∫Ω


≤

≤ max1, ((λ+ α)− (n/p))−1(n+ 2− α)v1− 1

pn ∥f∥Mλ

p (Rn) +

∫Ω

|f |dx. (1.26)

If Ω has finite measure, then the map Iα,Ω from Mλp (Ω) to B(Rn) defined

by

Iα,Ωf(x) ≡∫Ω

f(y)dy

|x− y|n−α∀x ∈ Rn,

for all f ∈Mλp (Ω) is linear and continuous.

Proof. By applying Lemma 1.32 with E = Ω, we deduce that∫Ω


≤∫

Ω∩Bn(x,1)


+

∫Ω\Bn(x,1)


≤

≤ mn (Bn(x, 1))1q− 1

p µwλ,q(n+ 2− α)v1− 1

qn ∥f∥Mλ

p (Rn) +

∫Ω\Bn(x,1)

|f |1n−α

dx ≤

≤ µwλ,q(n+ 2− α)v1− 1

qn ∥f∥Mλ

p (Rn) +

∫Ω

|f |dx,

for all x ∈ Rn. Hence, inequality (1.26) follows.

1.4 Sobolev spaces built on Morrey spaces

Definition 1.34. A domain Ω ⊂ Rn is called star-shaped with respect to the

point y ∈ Ω if for all x ∈ Ω the segment [x, y] ⊂ Ω. A domain Ω ⊂ Rn is

called star-shaped with respect to a point if for some y ∈ Ω it is star-shaped

with respect to the point y.

36


Definition 1.35. A domain Ω ⊂ Rn is called star-shaped with respect to the

ball B ⊂ Ω if for all y ∈ B and for all x ∈ Ω we have [x, y] ⊂ Ω. A domain

Ω ⊂ Rn is called star-shaped with respect to a ball if for some ball B ⊂ Ω it is

star-shaped respect to the ball B.

Definition 1.36. If 0 < d ≤ diamB ≤ diamΩ ≤ D, we set that Ω is star-

shaped with respect to a ball with the parameters d,D.

Definition 1.37. We call the set

Vx ≡ Vx,B =∪y∈B

(x, y)

a conic body with the vertex x constructed on the ball B.

A domain Ω star-shaped with respect to a ball B can be equivalently defined

in the following way: for all x ∈ Ω the conic body Vx ⊂ Ω.

Definition 1.38. If r, h ∈]0,+∞[, then we define the cone K(r, h) as follows

K(r, h) ≡(x′, xn) ∈ Rn : |x′| < rxn

h, xn < h

,

if n > 1, and

K(r, h) ≡]0, h[,

if n = 1.

We denote by O(n) the orthogonal group, i.e., the set of n× n matrices R

with real entries such that RRt = I = RtR.

Definition 1.39. Let Ω be an open subset of Rn.

(i) Let r, h ∈]0,+∞[. We say that Ω satisfies the cone property (or con-

dition) with parameters r, h provided that for each x ∈ Ω there exists

Rx ∈ O(n) such that

x+Rx(K(r, h)) ⊆ Ω.

(ii) We say that Ω satisfies the cone property (or condition), provided that

there exists r, h ∈]0,+∞[ such that Ω satisfies the cone property (or

condition) with parameters r, h.

37


Lemma 1.40. 1. A bounded open set Ω ⊂ Rn satisfies the cone condition if,

and only if, there exist s ∈ N and bounded domains Ωk, which are star-shaped

with respect to the balls Bk ⊂ Bk ⊂ Ωk, k = 1, . . . , s, such that Ω =s∪

k=1

Ωk.

2. An unbounded open set Ω ⊂ Rn satisfies the cone condition if, and only

if, there exist bounded domains Ωk, k ∈ N \ 0, which are star-shaped with

respect to the balls Bk ⊂ Bk ⊂ Ωk, k ∈ N \ 0, and are such that

1) Ω =∞∪k=1

Ωk,

2) 0 < infk∈N\0

diamBk ≤ supk∈N\0

diamΩk <∞,

3) the multiplicity of the covering ℵ(Ωk∞k=1) is finite.

(cf. e.g., Burenkov[13, Ch. 3.2])

Lemma 1.41. Let m0 ∈ N \ 0, 1 ≤ p1, . . . , pm0 , q ≤ ∞, λ ∈[0, n

pm

], for all

m = 1, . . . ,m0, 0 ≤ ν ≤ nq. Let Ω =

s∪k=1

Ωk, where s ∈ N\0 and Ωk ⊂ Rn are

bounded open sets. Furthermore, let fm, m = 1, . . . ,m0, and g be measurable

functions on Ω.

Suppose that there exists σm > 0 such that

∥g∥Mρ−νq (Ωk)

≤m0∑m=1

σm∥fm∥Mρ−λpm (Ωk)

, (1.27)

for all m = 1, . . . ,m0.

Then

∥g∥Mρ−νq (Ω)

≤ 2νs1q

m0∑m=1

σm∥fm∥Mρ−λpm (Ω)

. (1.28)

Proof. Let q <∞.

∥g∥qMρ−ν

q (Ω)= sup

x∈Ωρ>0

ρ−νq∥g∥qLq(B(x,ρ)∩Ω) =

= supx∈Ωρ>0

ρ−νq∫

B(x,ρ)∩Ω

|g(y)|qdy ≤ supx∈Ωρ>0

s∑k=1

ρ−νq∫

B(x,ρ)∩Ωk

|g(y)|qdy ≤

≤s∑

k=1

supx∈Ωρ>0

ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)≤

s∑k=1

supx∈Rnρ>0

ρ−νq∥g∥qLq(B(x,ρ)∩Ωk).

If x ∈ Ωk, then

ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)= sup

x∈Ωkρ>0

ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)= ∥g∥q

Mρ−νq (Ωk)

.

38


Let x ∈ Rn\Ωk. If B(x, ρ) ∩ Ωk = ∅, then ρ−ν∥g∥Lq(B(x,ρ)∩Ωk) = 0. Thus,

we can assume that B(x, ρ) ∩ Ωk = ∅.

Let ξ ∈ B(x, ρ) ∩ Ωk. By the triangle inequality, we have

B(x, ρ) ∩ Ωk ⊂ B(ξ, 2ρ) ∩ Ωk.

Hence,

ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)≤ 2νq sup

ξ∈Ωkρ>0

(2ρ)−νq∥g∥qLq(B(ξ,2ρ)∩Ωk)= 2νq∥g∥q

Mr−νq (Ωk)

.

By (1.27) and the Minkowski inequality it follows, that

∥g∥Mρ−νq (Ω)

≤ 2ν

(s∑

k=1

∥g∥qMρ−ν

q (Ωk)

) 1q

≤ 2ν

(s∑

k=1

(m0∑m=1

σm∥fm∥Mρ−λpm (Ωk)

)q) 1q

≤

≤ 2νm0∑m=1

(s∑

k=1

(σm∥fm∥Mρ−λ

pm (Ωk)

)q) 1q

= 2νm0∑m=1

σm

(s∑

k=1

∥fm∥qMρ−λ

pm (Ωk)

) 1q

.

Then(s∑

k=1

∥fm∥qMρ−λ

pm (Ωk)

) 1q

≤

(s∑

k=1

∥fm∥qMρ−λ

pm (Ω)

) 1q

= s1q ∥fm∥Mρ−λ

pm (Ω)

and inequality (1.28) follows.

The case in which some pm = ∞ is treated in a similar way with suprema

replacing sums.

The case q = ∞ is trivial and the statement holds for Ω =∪i∈I

Ωi, where I

is an arbitrary set of indices:

∥g∥Mν∞(Ω) = sup

i∈Isupx∈Ωρ>0

ρ−ν∥g∥L∞(B(x,ρ)∩Ωi) ≤m0∑m=1

σm∥fm∥Mλpm

(Ω).

Definition 1.42. Let Ω ⊂ Rn be an open set. Let l ∈ N, p ∈ [1,+∞] and

λ ∈[0, n

p

]. Then we define the Sobolev space of order l built on the Morrey

space Mλp (Ω), as the set

W l,λp (Ω) ≡

f ∈Mλ

p (Ω) : Dαwf ∈Mλ

p (Ω) ∀α ∈ Nn, |α| ≤ l,

where Dαwf is the weak derivative of f .

39


Then we set

∥f∥W l,λp (Ω) =

∑|α|≤l

∥Dαwf∥Mλ

p (Ω) ∀ f ∈ W l,λp (Ω).

In particular, W 0,λp (Ω) = Mλ

p (Ω) and W l,0p (Ω) = W l

p(Ω), where W lp(Ω)

denotes the classical Sobolev space of exponents l, p in Ω. It is obvious that

W l,λp (Ω) ⊂ W l

p(Ω).

Since Mλp (Ω) is a Banach space, one can exploit standard properties of the

weak derivatives and prove the following.

Theorem 1.43. Let Ω be an open subset of Rn. Let p ∈ [1,+∞] and

λ ∈[0, n

p

]. Then

(W l,λ

p (Ω), ∥ · ∥W l,λp (Ω)

)is a Banach space.

Then we have the following remark, which provides a family of equivalent

norms in W l,λp (Ω).

Remark 1.44. Let Ω be an open subset of Rn. Let l ∈ N, p ∈ [1,+∞] and

λ ∈[0, n

p

]. Let c(l) be the number of multi indexes α ∈ Nn such that |α| ≤ l.

Let q be a norm on Rc(l). Then the map qW l,λp (Ω) from W l,λ

p (Ω) to [0,+∞[

which takes u to qW l,λp (Ω)(u) ≡ q

((∥Dα

wu∥Mλp (Ω))|α|≤l

)is an equivalent norm in

W l,λp (Ω).

In particular,

( ∑|α|≤l

∥Dαwu∥tMλ

p (Ω)

) 1t

for t ∈ [1,+∞[ and max|α|≤l

∥Dαwu∥Mλ

p (Ω)

are all equivalent norms on W l,λp (Ω).

Definition 1.45. Let Ω ⊂ Rn be an open set. Let l ∈ N, p ∈ [1,+∞] and

λ ∈[0, n

p

]. Then we define the Sobolev space of order l built on the little

Morrey space Mλ,0p (Ω), as the set

W l,λ,0p (Ω) ≡

f ∈Mλ,0

p (Ω) : Dαwf ∈Mλ,0

p (Ω) ∀α ∈ Nn, |α| ≤ l.

Since Mλ,0p (Ω) is a closed subspace of Mλ

p (Ω), we can easily deduce the

validity of the following.

Theorem 1.46. Let Ω be an open subset of Rn. Let l ∈ N, p ∈ [1,+∞] and

λ ∈[0, n

p

]. Then W l,λ,0

p (Ω) is a closed proper subspace of W l,λp (Ω).

40


Proof. Let u ∈ W l,λp (Ω). Let ukk∈N be a sequence in W l,λ,0

p (Ω) which con-

verges to u in W l,λp (Ω). We want to show that u ∈ W l,λ,0

p (Ω).

Since uk → u in W l,λp (Ω) as k → ∞, we have

Dαwuk → Dα

wu ∀ |α| ≤ l in Mλp (Ω)

as k → ∞.

We know that Mλ,0p (Ω) is a closed subspace of Mλ

p (Ω). Therefore,

Dαwu ∈Mλ,0

p (Ω) ∀ |α| ≤ l,

and, thus, u ∈ W l,λ,0p (Ω).

Lemma 1.47. Let l ∈ N \ 0. Let m ∈ N, m < l. Let 1 ≤ p, q ≤ +∞,

0 ≤ λ ≤ np, 0 ≤ ν ≤ n

q. Suppose that for each bounded domain G ⊂ Rn

star-shaped with respect to a ball there exists c1 > 0 such that for each β ∈ Nn

satisfying |β| ≤ m and for all f ∈ W l,λp (G)

∥Dβwf∥Mν

q (G) ≤ c1∥f∥W l,λp (G).

Then for each open bounded set Ω ⊂ Rn satisfying the cone condition there

exists c2 > 0 such that

∥Dβwf∥Mν

q (Ω) ≤ c2∥f∥W l,λp (Ω)

for each β ∈ Nn satisfying |β| ≤ m and for all f ∈ W l,λp (Ω).

Proof. Let Ω satisfy the cone condition with the parameters r, h. By

Lemma 1.40

Ω =s∪

k=1

Ωk,

where s ∈ N and Ωk are bounded domains star-shaped with respect to the

balls Bk ⊂ Bk ⊂ Ωk.

Then

∥Dβwf∥Mν

q (Ωk) ≤ c1(Ωk)∥f∥W l,λp (Ωk)

, k = 1, . . . , s.

Hence, by Lemma 1.41

∥Dβwf∥Mν

q (Ω) ≤ 2νs1q maxk=1,...,s

c1(Ωk)∥f∥W l,λp (Ω).

41

Chapter 2

The embedding and

multiplication operators in

Sobolev Morrey spaces

2.1 The Sobolev Embedding Theorem

First we introduce the following notation.

Definition 2.1. Let p ∈ [1,+∞], l, n ∈ N\0, m ∈ N, m ≤ l,

λ, ν ∈ [0,+∞[. Let l + λ−m− ν = np. Then we set

q∗(l,m, n, p, λ, ν) ≡ n

(n/p)− (l + λ−m− ν).

If λ = ν = 0, then q∗(l,m, n, p, λ, ν) equals the classical Sobolev limiting ex-

ponent. If λ, ν ∈ [0,+∞[, then the exponent q∗(l,m, n, p, λ, ν) can be obtained

from the classical one by replacing l by l + λ and m by m+ ν.

We note that if l + λ − ν = np, then the equality which defines

q∗(l, 0, n, p, λ, ν) is equivalent to the equality

l =

(ν − n

q∗(l, 0, n, p, λ, ν)

)−(λ− n

p

).

We also note that

q∗(l, 0, n, p, λ, ν)

p> 1 whenever

l + λ > ν,

l + λ− ν < np,

43

2. The embedding and multiplication operators in Sobolev Morrey spaces

and

q∗(l −m, 0, n, p, λ, ν) = q∗(l,m, n, p, λ, ν).

Remark 2.2. Before we prove an analogue of the Sobolev Embedding Theorem,

we recall the following:

(i) Mλp (Rn) is continuously embedded into Mr−λ

p (Rn).

(ii) If Ω is a bounded domain, then we have Mr−λ

p (Ω) = Mλp (Ω) with equiv-

alent norms.

We are now ready to prove the following Sobolev Embedding Theorem.

Theorem 2.3. Let p ∈ [1,+∞[, l, n ∈ N\0, m ∈ N, m ≤ l, λ ∈[0, n

p

]. Let

Ω be a bounded open subset of Rn which satisfies the cone property. Then the

following statements hold.

(i) Let l−m+λ < np. Let ν ∈]λ, (l−m)+λ]. Then W l,λ

p (Ω) is continuously

embedded into Wm,νq∗(l,m,n,p,λ,ν)(Ω).

(ii) Let l−m+ λ < np. If p > 1, then W l,λ

p (Ω) is continuously embedded into

Wm,λq∗(l,m,n,p,λ,λ)(Ω).

(iii) Let l−m+λ > np. Then W l,λ

p (Ω) is continuously embedded into Wm∞(Ω).

Proof. (i) First let m = 0.

Let Ω be a bounded domain star-shaped with respect to the ball B =

B(x0, r), B ⊂ Ω. Then by Sobolev’s integral representation there existsM1 > 0

such that

|f(x)| ≤M1

∫B

|f |dy +∑|α|=l

∫Vx

(Dαwf)(y)

|x− y|n−ldy

for almost all x ∈ Ω for each (cf. e.g., Burenkov [13, Ch.3 p.112]).

Hence,

∥f∥Mr−ν

q∗(l,0,n,p,λ,ν)(Ω)

≤M1

∫B

|f |dy · ∥1∥Mr−ν

q∗(l,0,n,p,λ,ν)(Ω)

+

+∑|α|=l

∥∥∥∥∥∥∫Rn

Φα(y)

|x− y|n−ldy

∥∥∥∥∥∥Mr−ν

q∗(l,0,n,p,λ,ν)(Ω)

,

44


where Φα(y) =

Dαwf(y), if y ∈ Ω;

0, if y /∈ Ω.

Note that

∥1∥Mr−ν

q∗(l,0,n,p,λ,ν)(Ω)

= supx∈Ω

supρ>0

ρ−ν∥1∥Lq∗(l,0,n,p,λ,ν)(B(x,ρ)∩Ω) ≤

≤ supx∈Ω

max

sup

0<ρ≤(diamΩ)

υ1

q∗(l,0,n,p,λ,ν)n ρ−ν+

nq∗(l,0,n,p,λ,ν) , sup

ρ≥(diamΩ)

ρ−νmn(Ω)1

q∗(l,0,n,p,λ,ν)

=

= max

υ

1q∗(l,0,n,p,λ,ν)n mn(Ω)

−ν+ nq∗(l,0,n,p,λ,ν) ,mn(Ω)

−ν+ 1q∗(l,0,n,p,λ,ν)

<∞.

By Theorem 1.30 there exists c > 0 depending only on n, l, p,

q∗(l, 0, n, p, λ, ν) such that

∥∥∥∥∥∥∫Rn

Φα(y)

|x− y|n−ldy

∥∥∥∥∥∥Mr−ν

q∗(l,0,n,p,λ,ν)(Rn)

≤ c∥Φα∥Mr−λp (Rn)

≤

≤ 2λc∥Dαwf∥Mr−λ

p (Ω)≤ 2λc∥Dα

wf∥Mλp (Ω).

By Holder inequality∫B

|f |dy ≤ mn(B)1p′ ∥f∥Lp(Ω).

Therefore, there exist M2 > 0 and M3 > 0 such that

∥f∥Mνq∗(l,0,n,p,λ,ν)

(Ω) ≤ max1, (diamΩ)ν∥f∥Mr−ν

q∗(l,0,n,p,λ,ν)(Ω)

≤

≤ max1, (diamΩ)ν∥f∥Mr−ν

q∗(l,0,n,p,λ,ν)(Rn)

≤

≤M2

∥f∥Lp(Ω) +∑|α|=l

∥∥∥∥∥∥∫Rn

Φα(y)

|x− y|n−ldy

∥∥∥∥∥∥Mr−ν

q∗ (Rn)

≤

≤M3

∥f∥Mλp (Ω) +

∑|α|=l

∥Dαwf∥Mλ

p (Ω)

=M3∥f∥W l,λp (Ω), ∀ f ∈ W l,λ

p (Ω).

Hence, by Lemma 1.47, the statement of Theorem 2.3 follows.

Now let α : |α| = m. Then Dαwf ∈ W

l−|α|,λp (Ω) = W l−m,λ

p (Ω). Hence, there

45


exists a constant c1 > 0 such that

∥f∥Wm,νq∗(l,m,n,p,λ,ν)

(Ω) =∑|α|≤m

∥Dαwf∥Mν

q∗(l,m,n,p,λ,ν)(Ω) =

=∑|α|≤m

∥Dαwf∥Mν

q∗(l−m,0,n,p,λ,ν)(Ω) ≤ c1

∑|α|≤m

∥Dαwf∥W l−m,λ

p (Ω) ≤

≤ c1∑|α|≤m

∑|γ|≤l−m

∥Dγ+αw f∥Mλ

p (Ω) ≤

≤ c1∑|α|≤l

∥Dαwf∥Mλ

p (Ω) = c1∥f∥W l,λp (Ω), ∀ f ∈ W l,λ

p (Ω).

(ii) This case can be analized as case (i) by replacing q∗(l,m, n, p, λ, ν) by

q∗(l,m, n, p, λ, λ).

(iii) Now l − m + λ > np. Let Ω be a bounded domain star-shaped with

respect to the ball B = Bn(ξ, r0), B ⊂ Ω. Then by Sobolev’s integral repre-

sentation there exists c > 0 such that

|f(x)| ≤ c

∫Bn(ξ,r0)

|f |dx+∑|γ|=l

∫Vx

|(Dγwf)(y)|

|x− y|n−ldy

, (2.1)

for almost all x ∈ Ω and for all f ∈ W l,λp (Ω), and where Vx denotes the conical

body based on Bn(ξ, r0) and with vertex x (cf. e.g., Burenkov [13, Ch.3 p.112]).

We first consider case m = 0. So we now assume that l+λ > np. We plan to

estimate the supremum of |f | by exploiting inequality (2.1). Since∫

Bn(ξ,r0)

|f |dx

is a constant, it defines an element of C0b (Ω) ⊆ L∞(Ω). Next we prove that

the sum in the right hand side of (2.1) is bounded if f ∈ W l,λp (Ω).

We plan to treat separately case l < n and case l ≥ n.

Let l < n. Since l+λ > np, we can invoke Corollary 1.33 and conclude that

Il,Ω is linear and continuous from Mλp (Ω) to B(Rn).

Since Vx ⊆ Ω for all x ∈ Ω, we deduce that∣∣∣∣∣∣∫Vx

|h(y)||x− y|n−l

dy

∣∣∣∣∣∣ ≤ Il,Ω(|h|) ∀x ∈ Ω,

for all h ∈ Mλp (Ω). By the continuity of the restriction operator in Morrey

spaces and by the above mentioned continuity of Il,Ω, we deduce that the map

Jl,Ω from Mλp (Ω) to B(Ω) defined by

Jl,Ωh(x) ≡∫Vx

h(y)

|x− y|n−ldy ∀x ∈ Ω,

46


for all h ∈Mλp (Rn) satisfies the inequality

|J l,Ωh(x)| ≤ |Il,Ω(|h|)(x)| ≤ ∥Il,Ω∥L(Mλp (Ω),B(Rn))∥ |h| ∥Mλ

p (Ω) ≤

≤ ∥Il,Ω∥L(Mλp (Ω),B(Rn))∥h∥Mλ

p (Ω) ∀x ∈ Ω, (2.2)

for all h ∈Mλp (Ω). Then we deduce that

|f(x)| ≤ c

∫Bn(ξ,r0)

|f |dx+∑|γ|=l

|Jl,Ω[Dγwf ](x)|

≤

≤ c

[mn(Bn(ξ, r0))]1− 1

p∥f∥Lp(Ω) + ∥Il,Ω∥L(Mλp (Ω),B(Rn))

∑|γ|=l

∥Dγwf∥Mλ

p (Ω)

≤

≤ c([mn(Bn(ξ, r0))]

1− 1p + ∥Il,Ω∥L(Mλ

p (Ω),B(Rn))

)∥f∥W l,λ

p (Ω), (2.3)

for almost all x ∈ Ω and for all f ∈ W l,λp (Ω).

We now consider case l ≥ n. The embedding of Mλp (Ω) into Lp(Ω) and

inequality (2.1) and the Holder inequality imply that

|f(x)| ≤ c

∫Bn(ξ,r0)

|f |dx+∑|γ|=l

∫Ω

|(Dγwf)(y)|

|x− y|n−ldy

≤ (2.4)

≤ c

[mn(Bn(ξ, r0))]1− 1

p∥f∥Lp(Ω) +∑|γ|=l

∥Dγwf∥L1(Ω)(diamΩ)l−n

≤

≤ c

[mn(Bn(ξ, r0))]1− 1

p∥f∥Lp(Ω) + (diamΩ)l−n[mn(Ω)]1− 1

p

∑|γ|=l

∥Dγwf∥Lp(Ω)

≤

≤ c([mn(Bn(ξ, r0))]

1− 1p + (diamΩ)l−n[mn(Ω)]

1− 1p

)∥f∥W l,λ

p (Ω),

for almost all x ∈ Ω and for all f ∈ W l,λp (Ω). By Lemma 1.47, by the inequality

(2.3) for case l < n and by the inequality (2.4) for case l ≥ n, we deduce the

validity of statement (iii) in case m = 0.

Next we prove the statement (iii) in case m > 0. If f ∈ W l,λp (Ω), then

Dβwf ∈ W l−m,λ

p (Ω) for all |β| ≤ m. Now by assumption, we have (l−m)+λ > np.

Hence, case m = 0 with l replaced by l−m implies that W l−m,λp (Ω) ⊆ L∞(Ω)

and that there exists c1 > 0 such that

∥g∥L∞(Ω) ≤ c1∥g∥W l−m,λp (Ω) ∀ g ∈ W l−m,λ

p (Ω).

47


Hence, Dβwf ∈ L∞(Ω) for all β ∈ Nn such that |β| ≤ m, and

∥f∥Wm∞(Ω) ≤

∑|β|≤m

∥Dβwf∥L∞(Ω) ≤

≤ c1∑|β|≤m

∥Dβwf∥W l−m,λ

p (Ω) ≤ c1∑|β|≤m

∑|γ|≤l−m

∥Dγ+βw f∥Mλ

p (Ω) ≤

≤ c1

∑|β|≤m

∑|γ|≤l−m

1|γ+β|

∥f∥W l,λp (Ω)

for all f ∈ W l,λp (Ω). Hence, the proof of statement (iii) is complete.

2.2 Approximation by C∞ functions in


First we state a known Leibnitz formula for Sobolev spaces. For the proof one

can see for example [24, 5.2.3]

Theorem 2.4. Let l ∈ N. Let 1 ≤ p < +∞. Let Ω be a bounded open subset

of Rn. Let u ∈ W lp(Ω) and |α| ≤ l. If ζ ∈ C l

c(Ω), then ζu ∈ W lp(Ω) and

Dαw(ζu) =

∑β≤α

α

β

DβζDα−βw u, (2.5)

where

α

β

= α!β!(α−β)!

Proposition 2.5. Let Ω ⊂ Rn be an open set, l ∈ N. Let u, v ∈ Lloc1 (Ω). More-

over, assume that for any β ∈ Nn satisfying |β| ≤ l there exists 1 ≤ pβ < ∞

such that Dβwu ∈ Lloc

pβ(Ω) and Dγ

wv ∈ Llocp′β(Ω) for all γ ∈ Nn : |γ| ≤ l − |β|.

Then for any α ∈ Nn satisfying |α| ≤ l the weak derivative Dαw(uv) exists and

the Leibnitz formula holds:

Dαw(uv) =

∑0≤β≤α

α

β

DβwuD

α−βw v. (2.6)

Proof. Let u ∈ C∞(Ω) and v ∈ Lloc1 (Ω) be such that Dγ

wv ∈ Lloc1 (Ω). Then

Dαw(uv) =

∑0≤β≤α

α

β

DβuDα−βw v.

48

2.2 Approximation by C∞ functions in Sobolev Morrey spaces

Now let u be as in formulation, i.e. Dβwu ∈ Lloc

pβ(Ω). Let also

uk(x) = u(x) ∗ φ 1k(x) ∀ k ∈ N,

where φ 1k(x) as in definition 1.20 with t = 1

k.

Then, by Theorem 2.4, we have

Dαw(ukv) =

∑0≤β≤α

α

β

DβukDα−βw v =

=∑

0≤β≤α

α

β

Dβ[u ∗ φ 1k]Dα−β

w v.

Properties of mollifiers imply that

uk → u in Llocpβ(Ω),

Dβuk → Dβwu in Lloc

pβ(Ω),

as k → ∞. Thus,

Dαw(uv) =

∑0≤β≤α

α

β

DβwuD

α−βw v.

Theorem 2.6. Let l ∈ N. Let 1 ≤ p < +∞, 0 ≤ λ ≤ np. Let Ω be a

bounded open subset of Rn. Then for u ∈ W l,λ,0p (Ω) there exist functions um ∈

C∞(Ω) ∩W l,λp (Ω) such that

um → u in W l,λp (Ω).

Proof. We have Ω =∞∪i=1

Ωi, where

Ωi :=

x ∈ Ω: dist(x, ∂Ω) >

1

i

(i = 1, 2, . . .).

Write Vi := Ωi+3 − Ωi+1.

Choose also any open set V0 ⊂⊂ Ω so that Ω =∞∪i=0

Vi. Now let ζi∞i=1

be a smooth partition of unity subordinate to the open sets Vi∞i=0; that is,

suppose 0 ≤ ζi ≤ 1, ζi ∈ C∞c (Vi)∞∑i=0

ζi = 1 on Ω.

49


Next, choose any function u ∈ W l,λ,0p (Ω). By Theorem 2.4 we know that

Dαw(uζi) =

∑0≤β≤α

α

β

DβwuD

α−βw ζi.

Since Dβwu ∈ Mλ,0

p (Ω), Dα−βw ζi ∈ L∞(Ω), by Theorem 1.18 Dβ

wuDα−βw ζi ∈

Mλ,0p (Ω). Therefore, Dα

w(uζi) ∈ Mλ,0p (Ω) for all |α| ≤ l. Since W l,λ,0

p (Ω) is a

closed subspace of W l,λp (Ω), we have ζiu ∈ W l,λ,0

p (Rn) and supp (ζiu) ⊂ Vi.

Fix δ > 0. Choose then εi > 0 so small that ϕεi ∗ (ζiu) satisfies ∥ϕεi ∗ (ζiu)− ζiu∥W l,λp (Ω) ≤

δ2i+1 , (i = 0, 1, . . .)

supp [ϕεi ∗ (ζiu)] ⊂ Wi (i = 1, . . .),

for Wi := Ωi+4 − Ωi ⊃ Vi (i = 1, . . .).

Write v : =∞∑i=0

ϕεi ∗ (ζiu). This function belongs to C∞(Ω), since for each

open set V ⊂⊂ Ω there are at most finitely many nonzero terms in the sum.

Since u =∞∑i=0

ζiu, we have for each V ⊂⊂ Ω

∥v − u∥W l,λp (V ) ≤

∞∑i=0

∥ϕεi ∗ (ζiu)− ζiu∥W l,λp (Ω) ≤ δ

∞∑i=0

1

2i+1= δ.

Take the supremum over sets V ⊂⊂ Ω , to conclude ∥v − u∥W l,λp (Ω) ≤ δ.

2.3 Multiplication Theorems for Sobolev Mor-

rey spaces

Next we prove the following multiplication result which follows by the Holder

inequality.

Proposition 2.7. Let Ω be a bounded open subset of Rn. Let l ∈ N. Let

p, q, r ∈ [1,+∞] be such that 1r= 1

p+ 1

q. Let also 0 ≤ λ1 ≤ n

p, 0 ≤ λ2 ≤ n

q,

0 ≤ λ ≤ nr, λ ≡ λ1 + λ2. Then if u ∈ W l,λ1

p (Ω) and v ∈ W l,λ2q (Ω), we have

uv ∈ W l,λr (Ω).

Moreover, there exists c > 0 such that

∥uv∥W l,λr (Ω) ≤ c∥u∥

Wl,λ1p (Ω)

∥v∥W

l,λ2q (Ω)

∀ (u, v) ∈ W l,λ1p (Ω)×W l,λ2

q (Ω).

50

2.3 Multiplication Theorems for Sobolev Morrey spaces

Proof. By Proposition 2.5, we know that the Dαw weak derivative of u is de-

livered by the Leibnitz rule. Using Holder’s inequality for Morrey spaces, we

obtain

∥uv∥W l,λr (Ω) =

∑|α|≤l

∥Dαw(uv)∥Mλ

r (Ω) ≤

≤∑|α|≤l

∑|β|≤α

α

β

∥∥Dα−βw uDβ

wv∥∥Mλ

r (Ω)≤

≤∑|α|≤l

∑|β|≤α

α

β

∥∥Dα−βw u

∥∥M

λ1p (Ω)

∥∥Dβwv∥∥M

λ2q (Ω)

≤

≤ c∥u∥W

l,λ1p (Ω)

∥v∥W

l,λ2q (Ω)

.

We now extend a known multiplication result for Sobolev spaces of Zolesio

[61] (see also Valent [58], Runst and Sickel [51]).

Theorem 2.8. Let Ω be a bounded open subset of Rn. Let l ∈ N. Let p, q, r ∈

[1,+∞], and let 0 ≤ λ1 ≤ np, 0 ≤ λ2 ≤ n

q, 0 ≤ λ ≤ n

r. Assume that Ω has the

cone property, and p ≥ r, q ≥ r and

λ1 −n

p≥ λ− n

r, λ2 −

n

q≥ λ− n

r, (2.7)

l + λ1 + λ2 − λ

n>

1

p+

1

q− 1

r, (2.8)

l

n>

1

p+

1

q− 1

r. (2.9)

Then if u ∈ W l,λ1p (Ω) and v ∈ W l,λ2

q (Ω), we have uv ∈ W l,λr (Ω).

Moreover, there exists a positive number c such that

∥uv∥W l,λr (Ω) ≤ c∥u∥

Wl,λ1p (Ω)

∥v∥W

l,λ2q (Ω)

∀ (u, v) ∈ W l,λ1p (Ω)×W l,λ2

q (Ω). (2.10)

Proof. Let α ∈ Nn, |α| ≤ l. Then we set

Qα[u, v] =∑β≤α

α

β

Dα−βw uDβ

wv for all (u, v) ∈ W l,λ1p (Ω)×W l,λ2

q (Ω).

We want to prove that Qα defines a bilinear and continuous map from

W l,λ1p (Ω)×W l,λ2

q (Ω) to Mλr (Ω).

It is sufficies to show that if β ≤ α, |α| ≤ l, then

51


(A) the map from Wl−|α−β|,λ1p (Ω)×W

l−|β|,λ2q (Ω) to Mλ

r (Ω) which takes (u, v)

to uv is bilinear and continuous.

Before we perform the proof in several steps, we remark the following:

If either Wl−|α−β|,λ1p (Ω) or W

l−|β|,λ2q (Ω) is continuously embedded into

L∞(Ω) then (A) is true.

Indeed, if Wl−|α−β|,λ1p (Ω) → L∞(Ω), we know that

W l−|β|,λ2q (Ω) →Mλ2

q (Ω) →Mλr (Ω)

and the product

L∞(Ω)×Mλr (Ω) →Mλ

r (Ω)

is continuous.

Now we split the proof into several steps:

(a) maxl − |α− β|+ λ1 − n

p, l − |β|+ λ2 − n

q

> 0;

(b) maxl − |α− β|+ λ1 − n

p, l − |β|+ λ2 − n

q

< 0;

(c) maxl − |α− β|+ λ1 − n

p, l − |β|+ λ2 − n

q

= 0.

We begin by considering the case (a). Assume that

max

l − |α− β|+ λ1 −

n

p, l − |β|+ λ2 −

n

q

= l − |α− β|+ λ1 −

n

p> 0.

Then Theorem 2.3 (ii) implies that Wl−|α−β|,λ1p (Ω) is continuously embedded

into L∞(Ω). Thus, by remark above (A) follows.

If

max

l − |α− β|+ λ1 −

n

p, l − |β|+ λ2 −

n

q

= l − |β|+ λ2 −

n

q> 0,

then by similar way we see that Wl−|β|,λ2q (Ω) is continuously embedded into

L∞(Ω) and accordingly the truth of (A) follows.

Our next step will be proving of the case (b).

Here we distinguish the following cases (b1) l − |α− β| = 0;

(b2) l − |α− β| > 0.

(b3) l − |β| = 0;

(b4) l − |β| > 0.

52


Before we continue the proof we note that in case (b1)

l = |α− β| ≤ |α| ≤ l ⇔ |α| = l, |β| = 0, (2.11)

and in case (b3)

l = |β| ≤ |α| ≤ l ⇔ α = β.

So we can conclude that (b1) and (b3) cannot be both true.

We start from the case (b1)-(b4). Thus, we have Wl−|α−β|,λ1p (Ω) =

W 0,λ1p (Ω) =Mλ1

p (Ω) and by (2.11) Wl−|β|,λ2q (Ω) = W l,λ2

q (Ω).

We now choose ν ∈]λ2, λ2+l], then we set q1 =nq

n−q(l+λ2−ν) . Since λ2+l <nq,

Theorem 2.3(i) implies the following embedding

W l,λ2q (Ω) →M ν

q1(Ω)

Next we define r1 ∈ [1,+∞] such that 1r1

= 1p+ 1

q1.

Using Holder’s inequality for Morrey spaces, we get that the pointwise

product is bilinear and continuous from Mλ1p (Ω)×Mν

q1(Ω) to Mλ1+ν

r1(Ω).

Now we want to prove that Mλ1+νr1

(Ω) → Mλr (Ω). This embedding holds

provided the following two conditions.

1

r1=

1

p+

1

q1=

1

p+n− q(l + λ2 − ν)

nq=

1

p+

nq− (l + λ2 − ν)

n≤ 1

r,

and

(λ1 + ν)− n

r1≥ λ− n

r.

We can rewrite the second condition as follows

λ1 + ν

n− 1

r1≥ λ

n− 1

r,

λ1 + ν

n− 1

p−

nq− (l + λ2 − ν)

n≥ λ

n− 1

r,

l + λ1 + λ2 − λ

n≥ 1

p+

1

q− 1

r.

So we have the two conditions ln+ λ2−ν

n≥ 1

p+ 1

q− 1

r,

l+λ1+λ2−λn

≥ 1p+ 1

q− 1

r.

By assumptions (2.8), (2.9) we can choose ν sufficiently close to λ2 such

that the above inequalities hold as strict inequalities.

53


Therefore, by Theorem 1.15(iii) Mλ1+νr1

(Ω) is continuously embedded into

Mλr (Ω). Hence, uv ∈Mλ

r (Ω). Moreover, there exists c > 0 such that

∥uv∥Mλr (Ω) ≤ c∥u∥

Mλ1p (Ω)

∥v∥W

l,λ2q (Ω)

∀ (u, v) ∈Mλ1p (Ω)×W l,λ2

q (Ω).

Case (b2)-(b3) can be analyzed as case (b1)-(b4) by switching roles of

Wl−|α−β|,λ1p (Ω) and of W

l−|β|,λ2q (Ω).

Finally, we consider the case (b2)-(b4).

As above we choose γ ∈]λ1, λ1 + l] and µ ∈]λ2, λ2 + l]. Then we have the

following embeddings

W l−|α−β|,λ1p (Ω) →Mγ

p∗(Ω),

and

W l−|β|,λ2q (Ω) →Mµ

q∗(Ω),

where p∗ = npn−p(l−|α−β|+λ1−γ) , q

∗ = nqn−q(l−|β|+λ2−µ) .

Next we define r∗ such as 1r∗

= 1p∗

+ 1q∗.

Holder’s inequality for Morrey spaces implies that the pointwise product is

bilinear and continuous from Mγp∗(Ω)×Mµ

q∗(Ω) to Mγ+µr∗ (Ω).

Now if we recall Theorem 1.15(iii), we see that the embedding

Mγ+µr∗ (Ω) → Mλ

r (Ω) holds provided that

r ≤ r∗, (γ + µ)− n

r∗≥ λ− n

r. (2.12)

As before, we can rewrite this conditions as

1

r∗=

1

p∗+

1

q∗=

=n− p(l − |α− β|+ λ1 − γ)

np+n− q(l − |β|+ λ2 − µ)

nq=

=1

p+

1

q− l

n− l − |α|

n− λ1 − γ

n− λ2 − µ

n≤ 1

r,

l

n+l − |α|n

+λ1 − γ

n+λ2 − µ

n≥ 1

p+

1

q− 1

r,

l

n+l − |α|n

+λ1 − γ

n+λ2 − µ

n≥ l

n+λ1 − γ

n+λ2 − µ

n≥ 1

p+

1

q− 1

r.

54


and

(γ + µ)− n

r∗≥ λ− n

r,

γ + µ

n− 1

r∗≥ λ

n− 1

r,

γ + µ

n− 1

p− 1

q+l

n+l − |α|n

+λ1 − γ

n+λ2 − µ

n≥ λ

n− 1

r,

l + (l − |α|) + λ1 + λ2 − λ

n≥ 1

p+

1

q− 1

r,

l + (l − |α|) + λ1 + λ2 − λ

n≥ l + λ1 + λ2 − λ

n≥ 1

p+

1

q− 1

r.

As before, by assumptions (2.8), (2.9) we can choose γ and µ sufficiently

close to λ1 and λ2 respectively such that the above inequalities hold as strict

inequalities. Thus, we finish the case (b).

Next we will see case (c). We consider the following three separate cases

(c1) l − |α− β|+ λ1 − np< 0, l − |β|+ λ2 − n

q= 0;

(c2) l − |α− β|+ λ1 − np= 0, l − |β|+ λ2 − n

q< 0;

(c3) l − |α− β|+ λ1 − np= 0, l − |β|+ λ2 − n

q= 0.

We start with case (c1).

If l − |β| = 0, then λ2 = nqand W

l−|β|,λ2q (Ω) = L∞(Ω). Therefore, our

remark implies (A).

If l − |β| ≥ 1, we consider separately

(c11) q > 1;

(c12) λ2 > 0;

(c13) λ2 = 0, q = 1.

Let first l − |β| ≥ 1, q > 1. Then we choose q ∈]1, q[ so close to q so that

l + λ1 + λ2 − λ

n>

1

p+

1

q− 1

r,

l

n>

1

p+

1

q− 1

r.

Since q < q we have λ2 + l < nq, if l − |α− β| = 0,

λ2 + l − |β| < nq, if l − |α− β| ≥ 1.

55


But we have already proved such case (see cases (b1)-(b4) and (b2)-(b4)).

Thus (A) is true.

Now we see case (c12): l − |β| ≥ 1, λ2 > 0. Then we choose ν ∈]0, λ2[ so

close to λ2 so thatl + λ1 + ν − λ

n>

1

p+

1

q− 1

r.

Since ν < λ2 we have ν + l < nq, if l − |α− β| = 0,

ν + l − |β| < nq, if l − |α− β| ≥ 1.

But we have already proved such case (see cases (b1)-(b4) and (b2)-(b4)).

Thus (A) is true.

Finally we consider case (c13). Since λ2 = 0, q = 1, we have

W l−|β|,λ2q (Ω) = W

l−|β|1 (Ω).

Condition l − |β| = n implies that

Wl−|β|1 (Ω) → L∞(Ω).

Hence, by our remark (A) follows.

Case (c2) can be analyzed as case (c1) by switching roles of Wl−|α−β|,λ1p (Ω)

and of Wl−|β|,λ2q (Ω).

Now we consider case (c3).

If l − |α − β| = 0 or l − |β| = 0, then λ1 = npor λ2 = n

qrespectively

and Wl−|α−β|,λ1p (Ω) = L∞(Ω) or W

l−|β|,λ2q (Ω) = L∞(Ω). Therefore, our remark

implies (A).

Thus we can assume that

minl − |α− β|, l − |β| ≥ 1.

In such a case, we can split the proof of case (c3) into the following five

cases.

(c31) p > 1, q > 1;

(c32) p > 1, λ2 > 0;

(c33) λ1 > 0, q > 1;

(c34) λ1 > 0, λ2 > 0;

56


(c35) either (p, λ1) = (1, 0) or (q, λ2) = (1, 0).

In case (c31) we choose p ∈]1, p[ so close to p and q ∈]1, q[ so close to q so

thatl + λ1 + λ2 − λ

n>

1

p+

1

q− 1

r,

l

n>

1

p+

1

q− 1

r.

Since p < p, q < q we have λ1 + l − |α− β| < np

λ2 + l − |β| < nq.

But we have already proved such case (see case (b)). Thus (A) is true.

Now we see case (c32): p > 1, λ2 > 0. Then we choose p ∈]1, p[ so close to

p and ν ∈]0, λ2[ so close to λ2 so that

l + λ1 + ν − λ

n>

1

p+

1

q− 1

r,

l

n>

1

p+

1

q− 1

r.

Since p < p, ν < λ2 we have λ1 + l − |α− β| < np,

ν + l − |β| < nq.


Case (c33): λ1 > 0, q > 1. Then we choose γ ∈]0, λ1[ so close to λ1 and

q ∈]1, q[ so close to q so that

l + γ + λ2 − λ

n>

1

p+

1

q− 1

r,

l

n>

1

p+

1

q− 1

r.

Since q < q, γ < λ1 we have γ + l − |α− β| < np,

λ2 + l − |β| < nq.


Case (c34): λ1 > 0, λ2 > 0. We choose γ ∈]0, λ1[ so close to λ1 and

ν ∈]0, λ2[ so close to λ2 so that

l + γ + ν − λ

n>

1

p+

1

q− 1

r.

Since γ < λ1, ν < λ2 we have γ + l − |α− β| < np,

ν + l − |β| < nq.

57



Case (c35): either (p, λ1) = (1, 0) or (q, λ2) = (1, 0). By Sobolev embedding

theorem we have either

W l−|α−β|,λ1p (Ω) = W

l−|α−β|1 (Ω) → L∞(Ω).

or

W l−|β|,λ2q (Ω) = W

l−|β|1 (Ω) → L∞(Ω).

Hence, by our remark (A) follows.

We now show that if α ∈ Nn and |α| ≤ l, then

Dαw(uv) = Qα[u, v] ∀(u, v) ∈ W l,λ1

p (Ω)×W l,λ2q (Ω),

an equality which implies the validity of the Leibnitz rule. To do so, we take

p1 ∈ [1, p], p1 <∞, q1 ∈ [1, q], q1 <∞,

such that

p1 ≥ 1, 0− np1

≥ 0− n1,

q1 ≥ 1, 0− nq1

≥ 0− n1,

andl + 0 + 0− 0

n>

1

p1+

1

q1− 1

1.

Then we note that

W l,λ1p (Ω)×W l,λ2

q (Ω) → W lp(Ω)×W l

q(Ω) →W lp1(Ω)×W l

q1(Ω).

Let now uk and vk be sequences in C∞(Ω)∩W l

p1(Ω) and C∞(Ω)

∩W l

q1(Ω)

respectively such that uk converges to u in W lp1(Ω) and vk converges to v in

W lq1(Ω).

Next we observe that the pointwise product fromW lp1(Ω)×W l

q1(Ω) to L1(Ω)

is continuous.

Indeed, we know that uv = D0w(uv) = Q0[u, v] is continuous from

W lp1(Ω) × W l

q1(Ω) to L1(Ω). Since the pointwise product is bilinear and con-

tinuous from L1(Ω)× L∞(Ω) to L1(Ω) we conclude that∫Ω

ukvkDαφdx→

∫Ω

uvDαφdx.

58


Then ∫Ω

uvDαφdx = limk→∞

∫Ω

ukvkDαwφdx = lim

k→∞

∫Ω

Qα[uk, vk]φdx.

Since Qα is continuous from W lp1(Ω)×W l

q1(Ω) to L1(Ω), we have

limk→∞

∫Ω

Qα[uk, vk]φdx =

∫Ω

Qα[u, v]φdx,

and therefore ∫Ω

uvDαwφdx =

∫Ω

Qα[u, v]φdx.

Corollary 2.9. Let Ω be a bounded open subset of Rn which satisfies the cone

property. Let p, q ∈ [1,+∞]. Let l ∈ N\0. Let λ1 ∈ [0, n/p], λ2 ∈ [0, n/q],

p ≥ q, λ1 −n

p≥ λ2 −

n

q, (2.13)

and

l >n

p. (2.14)

Then the pointwise product from W l,λ1p (Ω) ×W l,λ2

q (Ω) to W l,λ2q (Ω) is bilinear

and continuous and the Leibnitz rule (2.6) holds.

Proof. It suffices to choose r = q, λ = λ2 in Theorem 2.8. We note that for

such a specific choice of the exponents, condition (2.9) implies the validity of

condition (2.8).

Then we have the following immediate consequence of the previous corol-

lary.


property. Let p ∈ [1,+∞]. Let l ∈ N\0. Let λ1 ∈ [0, n/p],

l >n

p. (2.15)

Then the pointwise product from W l,λ1p (Ω) ×W l,λ1

p (Ω) to W l,λ1p (Ω) is bilinear

and continuous and the Leibnitz rule (2.6) holds.

Proof. It suffices to choose p = q, λ2 = λ1 in the previous corollary.

59


We now prove in case l = 1 the following stronger result.

Proposition 2.11. Let Ω be a bounded open subset of Rn which satisfies the

cone property. Let p ∈ [1,+∞]. Let λ ∈ [0, n/p],

1 + λ >n

p. (2.16)

Then the pointwise product from W 1,λp (Ω)×W 1,λ

p (Ω) to W 1,λp (Ω) is bilinear and

continuous and the Leibnitz rule (2.6) holds.

Proof. We want to prove that if u, v ∈ W 1,λp (Ω), then uv ∈ W 1,λ

p (Ω).

To do so, we observe that ∀ j ∈ 1, . . . , n

(uv)xj= uxj

v + uvxj,

uxj∈Mλ

p (Ω), vxj∈Mλ

p (Ω).

Since 1 + λ > np, Theorem 2.3 (iii) implies that W 1,λ


embedded into L∞(Ω). Then by Theorem 1.18 the pointwise product is bilinear

and continuous from Mλp (Ω)×L∞(Ω) to M

λp (Ω) and from L∞(Ω)×Mλ

p (Ω) to

Mλp (Ω). Thus,

(uxjv) ∈Mλ

p (Ω), (uvxj) ∈Mλ

p (Ω),

and, therefore, (uv)xj∈Mλ

p (Ω) for all j ∈ 1, . . . , n.

60

Chapter 3

The composition operator in


3.1 Composition operator in Morrey spaces

Lemma 3.1. Let Ω be an open subset of Rn. Let Ω1 be a subset of R. Let

y ∈ Ω1. Let f be a Lipschitz continuous map from Ω1 to R. Let g ∈ M(Ω) be

such that g(x) ∈ Ω1 for almost all x ∈ Ω. Then

|f(g(x))| ≤ Lip(f)|g(x)|+ Lip(f)|y|+ |f(y)| (3.1)

for almost all x ∈ Ω.

Proof.

|f(g(x))| ≤ |f(g(x))− f(y)|+ |f(y)| ≤ Lip(f)|g(x)− y|+ |f(y)| ≤

≤ Lip(f)|g(x)|+ Lip(f)|y|+ |f(y)|

for almost all x ∈ Ω.

Lemma 3.2. Let Ω be a bounded open subset of Rn. Let p ∈ [1,+∞]. Let

λ ∈[0, n

p

]. Then 1 ∈Mλ

p (Ω).

Proof. If p = +∞, then we have λ = 0 and accordingly Mλp (Ω) = Lp(Ω) =

L∞(Ω) and 1 ∈ L∞(Ω) =Mλp (Ω).

Now let p ∈ [1,+∞[. Then we have

wλ(r)∥1∥Lp(Bn(x,r)∩Ω) ≤ wλ(r) (mn(Bn(x, r) ∩ Ω))1p

61

3. The composition operator in Sobolev Morrey spaces

for all x ∈ Ω and r ∈]0,+∞[. Hence,

wλ(r)∥1∥Lp(Bn(x,r)∩Ω) ≤ v1pn r

np−λ ≤ v

1pn

for all x ∈ Ω and r ∈]0, 1] and

wλ(r)∥1∥Lp(Bn(x,r)∩Ω) ≤ (mn(Ω))1p

for all x ∈ Ω and r ∈]1,+∞[ and accordingly 1 ∈Mλp (Ω) and

∥1∥Mλp (Ω) ≤ sup

v

1pn , (mn(Ω))

1p

.

Now we consider the case of Morrey spaces and we introduce the following

sufficient condition.

Proposition 3.3. Let Ω be a bounded open subset of Rn. Let Ω1 be a subset

of R. Let y ∈ Ω1. Let p ∈ [1,+∞]. Let λ ∈[0, n

p

]. Let g ∈ Mλ

p (Ω) be such

that g(x) ∈ Ω1 for almost all x ∈ Ω. Let f be a measurable function from Ω1

to R. Assume that there exists a, b > 0 such that

|f(y)| ≤ a|y|+ b, y ∈ Ω1. (3.2)

Then f g ∈Mλp (Ω) and for any y ∈ Ω1

∥f g∥Mλp (Ω) ≤ a∥g∥Mλ

p (Ω) + b∥1∥Mλp (Ω). (3.3)

Proof.

∥f g∥Mλp (Ω) ≤ ∥a|g|+ b∥Mλ

p (Ω) ≤ a∥g∥Mλp (Ω) + b∥1∥Mλ

p (Ω).

Remark 3.4. If f is a Lipschitz continuous function on Ω1, then Lemma 3.1

implies that condition (3.2) is satisfied with a = Lip(f), b = Lip(f)|y|+ |f(y)|.

Hence, by Proposition 3.3 for any y ∈ Ω1

∥f g∥Mλp (Ω) ≤ Lip(f)∥g∥Mλ

p (Ω) + ∥1∥Mλp (Ω)(Lip(f)|y|+ |f(y)). (3.4)

62

3.2 Composition operator in Sobolev Morrey spaces

Corollary 3.5. Let conditions of Proposition 3.3 are satisfied. If also 0 ∈ Ω1

and f is a Lipschitz continuous function on Ω1, then


p (Ω) + |f(0)| ·∥1∥Mλp (Ω).

Corollary 3.6. Let conditions of Proposition 3.3 are satisfied. If also 0 ∈ Ω1,

f(0) = 0 and f is a Lipschitz continuous function on Ω1, then


p (Ω).

Corollary 3.7. Let Ω be a bounded open subset of Rn. Let p ∈ [1,+∞]. Let

λ ∈[0, n

p

]. Let f be a locally Lipschitz continuous function from R to itself.

Then

Tf [Mλp (Ω) ∩ L∞(Ω)] ⊆Mλ

p (Ω) ∩ L∞(Ω).

(Note that in general Mλp (Ω) * L∞(Ω)).

Proof. Let g ∈Mλp (Ω)∩L∞(Ω). We set Ω1 =

[−∥g∥L∞(Ω), ∥g∥L∞(Ω)

]. Since Ω1

is a finite segment, f is Lipschitz continuous on Ω1. Hence, by Corollary 3.5

∥f g∥Mλp (Ω) < +∞.

We also have

∥f g∥L∞(Ω) ≤ ∥f∥L∞(Ω1) < +∞.

So, f g ∈Mλp (Ω) ∩ L∞(Ω).

3.2 Composition operator in Sobolev Morrey

spaces

Next we try to understand whether the Lipschitz continuity of a function f of

a real variable is enough to ensure that Tf [W1,λp (Ω)] ⊆ W 1,λ

p (Ω) under suitable

conditions on the exponents. To do so, we face the problem of taking the

distributional derivatives of the composite function f g, and we expect to

prove that

Dj(f g) = (f ′ g)Djg ∀ j ∈ 1, . . . , n.

63


However, it is not clear what f ′g should mean. Indeed, f ′ is defined only up to

a set of measure zero Nf and g←(Nf ) may have a positive measure, and even fill

the whole of Ω, and (f ′ g)(x) makes no sense when x ∈ g←(Nf ). Classically,

one circumvents such a difficulty by introducing a result of de la Vallee-Poussin

which states that both Dj(f g) and Djg vanish on g←(Nf ). Accordingly, it

suffices to define (f ′ g)(x) when x ∈ Ω\ g←(Nf ), and to replace (f ′ g)(x) by

0 in g←(Nf ). We find convenient to introduce a symbol for the function which

equals (f ′ g)(x) when x ∈ Ω \ g←(Nf ) and 0 elsewhere. Then we introduce

the following.

Definition 3.8. Let Ω be an open subset of Rn. Let Ω1 be a measurable

subset of R. Let g be a measurable function from Ω to R. Let the set Ng ≡

x ∈ Ω: g(x) /∈ Ω1 have measure zero.

Let H be a Borel subset of Ω1 of measure zero. Let h be a Borel measurable

function from Ω \H to R. Let hg be the function from Ω to R defined by

hg ≡

0, if x ∈ g←(H) ∪Ng,

(h g)(x), if x ∈ Ω \ (g←(H) ∪Ng).(3.5)

By definition, the function hg is measurable. Next we note that the fol-

lowing holds.

Lemma 3.9. Let Ω, Ω1, h, H be as in Definition 3.8. Let g, g1 be a measurable

functions from Ω to R such that g(x), g1(x) ∈ Ω1 for almost all x ∈ Ω. If

g(x) = g1(x) for almost all x ∈ Ω, then (hg)(x) = (hg1)(x) for almost all

x ∈ Ω.

Proof. Let N be a measurable subset of measure zero of Ω such that

g(x) = g1(x) and g(x), g1(x) ∈ Ω1 for all x ∈ Ω \ N . Since N has mea-

sure zero, it suffices to show that (hg)(x) = (hg1)(x) for all x ∈ Ω \N .

If x ∈ (Ω\N)∩g←(H), then g1(x) = g(x) ∈ H and x ∈ (Ω\N)∩g←(H), and

accordingly (hg1)(x) = 0 = (hg)(x). If instead x ∈ (Ω \N) ∩ (Ω \ g←(H)),

then g1(x) = g(x) /∈ H and accordingly x ∈ (Ω \ N) ∩ (Ω \ g←(H)) and

(hg1)(x) = (h g1)(x) = (h g)(x) = (hg)(x). Hence, (hg1)(x) = (hg)(x)

for all x ∈ Ω \N .

By the previous Lemma, it makes sense to introduce the following.

64


Definition 3.10. Let Ω, Ω1, h, H be as in Definition 3.8. If G is an equiv-

alence class of measurable functions g from Ω to R such that g(x) ∈ Ω1 for

almost all x ∈ Ω, then we define hG to be the equivalence class of measurable

functions from Ω to R which are equal to hg almost everywhere for at least a

g ∈ G.

If Ω be an open subset of Rn. We say that g ∈ Lloc1 (Ω) is zero on a subset A

of Ω provided that g(x) = 0 for almost all x ∈ A, for at least a representative

g of g (and thus for all representatives of g).

Remark 3.11. Let g1, g2 ∈ Lloc1 (Ω). Let g1 = g2 almost everywhere in Ω. If

A is a subset of R, then the symmetric difference g←1 (A)g←2 (A) has measure

zero. Indeed, g←1 (A)g←2 (A) ⊆ x ∈ Ω: g1(x) = g2(x).

Then we have the following n dimensional form of a result of de la Vallee-

Poussin [21]. For a proof, we refer to Marcus and Mizel [34].

Theorem 3.12. Let Ω be an open subset of Rn. Let g ∈ W 1,loc1 (Ω) and if N

is a subset of R of measure zero, then (D1g, . . . , Dng) = 0 on g←(N) for any

representative g of g.

Then we introduce the following form of the chain rule (see Marcus and

Mizel [34]).

Proposition 3.13. Let Ω be an open subset of Rn. Let Ω1 be an interval of

R. Let f be Lipschitz continuous function from Ω1 to R. Let

W 1,loc1 (Ω,Ω1) ≡ g ∈ W 1,loc

1 (Ω): g(x) ∈ Ω1 for almost all x ∈ Ω

for all representatives g of g.

Let Nf be the subset of Ω1 such that Ω1 \ Nf is the set of points of Ω1 where

f is differentiable. Let g ∈ W 1,loc1 (Ω,Ω1). Let f ′g be defined as in Definition

3.10 (with h = f ′, H = Nf). Then the chain rule

Dj(f g) = (f ′g)Djg, (3.6)

holds in the sense of distributions in Ω for all j ∈ 1, . . . , n.

65


Then we introduce the following sufficient condition for Sobolev Morrey

spaces of order one.

Proposition 3.14. Let Ω be a bounded open subset of Rn which satisfies the

cone property. Let Ω1 be an interval of R. Let p ∈ [1,+∞]. Let λ ∈[0, n

p

].

Let f be Lipschitz continuous function from Ω1 to R. Let

W 1,λp (Ω,Ω1) ≡ g ∈ W 1,λ

p (Ω) : g(x) ∈ Ω1 for almost all x ∈ Ω

for all representatives g of g.

Then

Tf [W1,λp (Ω,Ω1)] ⊆ W 1,λ

p (Ω).

Let Nf be the subset of Ω1 such that Ω1 \Nf is the set of points of Ω1 where

f is differentiable. Let g ∈ W 1,λp (Ω,Ω1). Let f ′g be defined as in Definition

3.10 (with h = f ′, H = Nf). Then f ′g ∈ L∞(Ω) and the chain rule formula

(3.6) holds in the sense of distributions in Ω for all j ∈ 1, . . . , n. Moreover,

∥f g∥W 1,λp (Ω) ≤

≤ (Lip(f)|y|+ |f(y)|) + Lip(f)(∥g∥W 1,λp (Ω) + ∥1∥Mλ

p (Ω)), (3.7)

for all g ∈ W 1,λp (Ω,Ω1) and for all y ∈ Ω1.

Proof. By Remark 3.4, we know that inequality (3.4) holds for all

g ∈ W 1,λp (Ω,Ω1) ⊆ Mλ

p (Ω) and for all y ∈ Ω1.

Now let g ∈ W 1,λp (Ω,Ω1). Let g be a representative of g. Let Ng be a subset

of measure 0 of Ω such that

g ∈ Ω1 ∀ x ∈ Ω \Ng.

If x ∈ Ω \ (Ng ∪ g←(Nf )), then

|f ′(g(x))| =∣∣∣∣ limη→g(x)

f(g(x))− f(η)

g(x)− η

∣∣∣∣ ≤ Lip(f).

Since f ′g = 0 for all x ∈ g←(Nf ), we conclude that

|(f ′g)(x)| ≤ Lip(f) a.e. in Ω.

66


and accordingly that f ′g ∈ L∞(Ω) and that ∥f ′g∥L∞(Ω) ≤ Lip(f) < +∞.

Then by the multiplication Theorem 1.18 and by the membership of Djg in

Mλp (Ω), we have (f ′g)Djg ∈Mλ

p (Ω) and

∥(f ′ g)Djg∥Mλp (Ω) ≤ Lip(f)∥Djg∥Mλ

p (Ω) (3.8)

for all j ∈ 1, . . . , n. Thus the right hand side of equality (3.6) belongs to

Mλp (Ω) for all g ∈ W 1,λ

p (Ω).

By the formula (3.6) for the chain rule, the inequalities (3.3), (3.8) imply

that

∥f g∥W 1,λp (Ω) = ∥f g∥Mλ

p (Ω) +n∑

j=1

∥(f ′g)Djg∥Mλp (Ω) ≤

≤ Lip(f)∥g∥Mλp (Ω) + ∥1∥Mλ

p (Ω)(Lip(f)|y|+ |f(y)|) +n∑

j=1

Lip(f)∥Djg∥Mλp (Ω) ≤

≤ (Lip(f)|y|+ |f(y)|) + Lip(f)(∥g∥W 1,λp (Ω) + ∥1∥Mλ

p (Ω)), (3.9)

for all g ∈ W 1,λp (Ω) and y ∈ Ω1, and thus inequality (3.7) holds true.


property. Let p ∈ [1,+∞]. Let λ ∈[0, n

p

]. Let f be a function from R to itself.

Then the following statements hold.

(i) If (1+λ) > npand if f is locally Lipschitz continuous, then Tf [W

1,λp (Ω)] ⊆

W 1,λp (Ω).

(ii) If (1 + λ) ≤ npand if f is Lipschitz continuous, then Tf [W

1,λp (Ω)] ⊆

W 1,λp (Ω).

Proof. We first consider statement (i). The Sobolev Embedding Theorem im-

plies thatW 1,λp (Ω) is continuously embedded into L∞(Ω). Thus if g ∈ W 1,λ

p (Ω),

there exists a bounded subset Ω1 of R such that g(x) ∈ Ω1 for almost all

x ∈ Ω. Since f∣∣Ω1

is Lipschitz continuous, Proposition 3.14 implies that

f g ∈ W 1,λp (Ω).

Statement (ii) is an immediate consequence of Proposition 3.14 with

Ω1 = R.

67


We summarize in the following statement some facts we need in the sequel

in case (1 + λ) > npand which are immediate consequence of Proposition 2.11

and Proposition 3.14.

Corollary 3.16. Let p ∈ [1,+∞], λ ∈[0, n

p

], (1+λ) > n

p. Let Ω be a bounded

open subset of Rn which satisfies the cone property. Let Ω1 be a bounded open

interval of R. Then the following statements hold.

(i) W 1,λp (Ω) is a Banach algebra.

(ii) If (f, g) ∈ C0,1(Ω1)×W 1,λp (Ω,Ω1), then f g ∈ W 1,λ

p (Ω). Moreover, there

exists an increasing function ψ from [0,+∞[ to itself such that

∥f g∥W 1,λp (Ω) ≤ ∥f∥C0,1(Ω1)ψ(∥g∥W 1,λ

p (Ω)) (3.10)

for all (f, g) ∈ C0,1(Ω1)×W 1,λp (Ω,Ω1).

3.3 Continuity of the composition operator in


Corollary 3.16 shows that if (1 + λ) > npthe composition T maps C0,1(Ω1) ×

W 1,λp (Ω,Ω1) to W

1,λp (Ω). Now we want to understand for which f ’s the com-

position operator Tf is continuous in W 1,λp (Ω,Ω1). By following [31],[30], the

idea is that if f is a polynomial, then Tf is continuous in W 1,λp (Ω). Indeed,

for (1 + λ) > np, the space W 1,λ

p (Ω) is a Banach algebra. Then we exploit

inequality (3.10) to show that if f is a limit of polynomials, then Tf is contin-

uous. Actually, such a scheme can be applied in a somewhat abstract general

setting, which we now introduce. Let X be a Banach algebra with unity. Let

m ∈ N\0. In the applications of the present notes we are interested in the

specific case m = 1, but here we present a more general case, which can be

applied to analyze vector valued functions of Sobolev Morrey.

We first note that if p belongs to the space P(Rm) of polynomials in m

real variables with real coefficients, then it makes perfectly sense to compose

68

3.3 Continuity of the composition operator in Sobolev Morrey spaces

p with some x ≡ (x1, . . . , xm) ∈ Xm. Namely, if p is defined by the equality

p(ξ1, . . . , ξm) ≡∑|η|≤deg p, η∈Nm

aηξη11 . . . ξηmm , with aη ∈ R, (ξ1, . . . , ξm) ∈ Rm, (3.11)

then we set

τp[x] ≡∑

|η|≤deg p, η∈Nm

aηxη11 . . . xηmm , ∀x ≡ (x1, . . . , xm) ∈ Xm, (3.12)

where the product between the xj’s is that of X , and where we understand

that x0 is the unit element of X , for all x ∈ X . Next we state the following

result of [30, Thm. 3.1].

Theorem 3.17. Let ∥ · ∥Y be a norm on P(Rm). Let Y be the completion of

P(Rm) with respect to the norm ∥ · ∥Y . Let X be a real commutative Banach

algebra with unity. Let X be a real Banach space. Assume that there exists a

linear continuous and injective map J of X into X . Let A be a subset of Xm.

Assume that there exists an increasing function ψ of [0,+∞) to itself such that

∥J [p(x1, . . . , xm)]∥X ≤ ∥p∥Y ψ (∥(x1, . . . , xm)∥Xm) , (3.13)

for all (p, (x1, . . . , xm)) ∈ P(Rm) × A. Then there exists a unique map A of

Y ×A to X such that the following two conditions hold

(i) A[p, x] = J [p(x)], for all (p, x) ∈ P(Rm)×A.

(ii) For all fixed x ≡ (x1, . . . , xm) ∈ A, the map y 7−→ A[y, x] is continuous

from Y to X .

Furthermore, the map A[·, x] of (ii) is linear, and A is continuous from Y×

A to X , and if y ∈ Y, y = limj→∞ pj in Y, pj ∈ P(Rm), x ≡ (x1, . . . , xm) ∈ A,

then

(iii) A[y, x] = limj→∞ J [pj(x)] in X ;

(iv) ∥A[y, x]∥X ≤ ∥y∥Y ψ (∥x∥Xm).

We shall call A[y, x] the ’abstract’ composition of y and x.

69


We now turn to apply the above theorem to the case of Sobolev Morrey

spaces. To do so, we need the following.

Proposition 3.18. Let Ω1 be a nonempty bounded open interval of R. Then

C1(Ω1) is a completion of the space (P(R), ∥ · ∥C0,1(Ω1)).

Proof. We first note that

supΩ1

|f ′| = Lip(f) ∀ f ∈ C1(Ω1).

Then we have

∥f∥C1(Ω1) = ∥f∥C0(Ω1) + ∥f ′∥C0(Ω1) = ∥f∥C0(Ω1) + Lip(f) = ∥f∥C0,1(Ω1)

for all f ∈ C1(Ω1). Hence,

∥f∥C0,1(Ω1) = ∥f∥C1(Ω1) ∀ p ∈ P(R).

Since C1(Ω1) is a Banach space and the restriction map from P(R) to C1(Ω1)

which takes p ∈ P(R) which takes p to p∣∣Ω1

is a linear isometry from (P(R), ∥ ·

∥C1(Ω1)) to(p∣∣Ω1

: p ∈ P(R), ∥ · ∥C1(Ω1)

)it suffices to show that p

∣∣Ω1

: p ∈

P(R) is dense in C1(Ω1). Let f ∈ C1(Ω1). Let x0 ∈ Ω1. Then

f(x) = f(x0) +

x∫x0

f ′(t)dt ∀x ∈ Ω1.

Now by the Weierstrass approximation Theorem, there exists a sequence

qjj∈N in P(R) such that

limj→N

qj = f ′ uniformly in Ω1.

Then if we set

pj(x) ≡ f(x0) +

x∫x0

qj(t)dt ∀x ∈ R,

for all j ∈ N, we have

∥f − pj∥C1(Ω1) ≤ supx∈Ω1

∣∣∣∣∣∣x∫

x0

(f ′ − qj)dt

∣∣∣∣∣∣+ ∥f ′ − qj∥C0(Ω1) ≤

≤ m1(Ω1)∥f ′ − qj∥C0(Ω1) + ∥f ′ − qj∥C0(Ω1) ≤

≤ (1 +m1(Ω1))∥f ′ − qj∥C0(Ω1) ∀ j ∈ N,

and accordingly limj→∞

∥f − pj∥C1(Ω1) = 0.

70

3.4 Lipschitz continuity of the composition operator in Sobolev Morrey spaces

Then by applying Theorem 3.17, we obtain the following.

Theorem 3.19. Let p ∈ [1,+∞], λ ∈[0, n

p

], (1 + λ) > n

p. Let Ω be a

bounded open subset of Rn which satisfies the cone property. Let Ω1 be a

bounded open interval of R. Then the composition operator T is continuous

from C1(Ω1)×W 1,λp (Ω,Ω1) to W

1,λp (Ω).

Proof. We set ∥ · ∥Y = ∥ · ∥C0,1(Ω1), X = X = W 1,λp (Ω), A = W l,λ

p (Ω,Ω1), J

equal to the identity map, m = 1. As we have shown, C1(Ω1) is a completion

of (P(R), ∥ · ∥C0,1(Ω1)). By Corollary 3.16, W 1,λp (Ω) is a Banach algebra and

there exists a function ψ as in (3.10). Then by Theorem 3.17, there exists a

unique map A from C1(Ω1)×W 1,λp (Ω,Ω1) to W

1,λp (Ω) such that the following

two conditions hold

(i) A[p, g] = τp[g] for all (p, g) ∈ P(R)×A.

(ii) For each fixed g ∈ W 1,λp (Ω,Ω1), the map from C1(Ω1) to W

1,λp (Ω) which

takes f to f g is continuous.

Moreover, A is continuous. Clearly, T satisfies (i), and inequality (3.10) implies

that T satisfies (ii). Hence, we must necessarily have

A[f, g] = T [f, g] ∀(f, g) ∈ C1(Ω1)×W 1,λp (Ω,Ω1).

As a consequence, T is continuous on C1(Ω1)×W 1,λp (Ω,Ω1).

3.4 Lipschitz continuity of the composition

operator in Sobolev Morrey spaces

Next we prove a Lipschitz continuity statement for the composition opera-

tor. For related results in Besov spaces, we refer to Bourdaud and Lanza de

Cristoforis [9].

Theorem 3.20. Let p ∈ [1,+∞], λ ∈[0, n

p

], (1+λ) > n


open subset of Rn which satisfies the cone property. If f ∈ C1,1loc (R), then Tf

maps W 1,λp (Ω) to itself and Lipschitz continuous on the bounded subsets of

W 1,λp (Ω).

71


Proof. Let B be a bounded subset of W 1,λp (Ω). Since W 1,λ


embedded into L∞(Ω), the set B is a bounded subset of L∞(Ω) and there exists

a closed interval B of R such that

[−∥g∥L∞(Ω), ∥g∥L∞(Ω)

]⊆ B ∀ g ∈ B.

Now let g1, g2 ∈ B. Since f is continuously differentiable, we can write

(f g2)(x)− (f g1)(x) =

=

1∫0

f ′[g1(x) + t(g2(x)− g1(x))](g2(x)− g1(x))dt ∀ x ∈ Ω.

Next we fix x ∈ Ω, r ∈]0,+∞[. By the Minkowski inequality for integrals, we

have

wλ(r)∥f g2 − f g1∥Lp(Ω∩Bn(x,r)) ≤

≤1∫

0

wλ(r)∥f ′[g1(·) + t(g2(·)− g1(·))](g2(·)− g1(·))∥Lp(Ω∩Bn(x,r)) ≤

≤1∫

0

wλ(r) supB

|f ′| ∥g2 − g1∥Lp(Ω∩Bn(x,r)) ≤

≤ supB

|f ′| ∥g2 − g1∥Mλp (Ω).

Hence,

∥f g2 − f g1∥Mλp (Ω) ≤ sup

B|f ′| ∥g2 − g1∥Mλ

p (Ω). (3.14)

Next we fix j ∈ 1, . . . , n and we try to estimate

∥(Dj)w f g2 − f g1∥Mλp (Ω) = (3.15)

= ∥f ′(g2)(Dj)w g2 − f ′(g1)(Dj)w g1∥Mλp (Ω) ≤

≤ ∥f ′ g2 − f ′ g1∥L∞(Ω) ∥(Dj)w g2∥Mλp (Ω)+

+∥f ′ g1∥L∞(Ω) ∥(Dj)w g2 − (Dj)w g1∥Mλp (Ω) ≤

≤ Lip(f ′∣∣B)∥g2 − g1∥L∞(Ω) sup

g∈B∥g∥W 1,λ

p (Ω) + supB

|f ′| ∥g2 − g1∥W 1,λp (Ω) ≤

≤Lip(f ′

∣∣B)∥I∥L(W 1,λ

p (Ω),L∞(Ω)) supg∈B

∥g∥W 1,λp (Ω) + sup

B|f ′|∥g2 − g1∥W 1,λ

p (Ω).

72

3.5 Differentiability properties of the composition operator in Sobolev Morreyspaces

By inequalities (3.14) and (3.15), we conclude that

∥f g2 − f g1∥W 1,λp (Ω) ≤

(1 + n) sup

B|f ′|+

+nLip(f ′∣∣B)∥I∥L(W 1,λ

p (Ω),L∞(Ω)) supg∈B

∥g∥W 1,λp (Ω)

∥g2 − g1∥W 1,λ

p (Ω).

3.5 Differentiability properties of the compo-

sition operator in Sobolev Morrey spaces

Next we turn to the question of differentiability, and by following [30], we note

that the following holds.

Lemma 3.21. Let X be a commutative real Banach algebra with unity 1X .

Let P(Rm) be the set of real polynomials in m real variables. Let p ∈ P(Rm)

be defined by

p(η) ≡∑

|γ|≤deg p

aγxγ11 . . . xγmm , ∀ η ≡ (η1, . . . , ηm) ∈ Rm.

The map τp of Xm to X defined by setting

τp[x1, . . . , xm] ≡∑

|γ|≤deg p

aγxη11 . . . xηmm , ∀ (x1, . . . , xm) ∈ Xm,

with the understanding that x0 ≡ 1X , for all x ∈ X , is of class Cr(Xm,X ), for

all r ∈ N ∪ ∞. Furthermore, the differential of τp [·] at x♯ ≡ (x♯1, . . . , x♯m) is

delivered by the map

Xm ∋ (h1, . . . , hm) 7→m∑i=1

τ ∂p∂xi

[x♯] ∗ hi ∈ X .

Once more, we plan to proceed by approximation and show that Tf is of

class Cr if f is a limit of polynomials with an appropriate norm. As we shall

see, it turns out that a right choice for the norm is the following

∥p∥Yr =∑

|γ|≤r, γ∈Nm

∥Dγp∥Y , ∀p ∈ P(Rm). (3.16)

Then we define Yr to be the completion of the space (P(Rm), ∥·∥Yr). As is well

known, Yr is unique up to a linear isometry, and we always choose Yr ⊆ Y .

Then we have the following obvious

73


Remark 3.22. If r, s ∈ N, s ≤ r, then

Yr ⊆ Ys, ∥y∥Ys ≤ ∥y∥Yr , ∀ y ∈ Yr.

Now we have the following (cf. Lanza de Cristoforis [30, Thm. 2.4]).

Theorem 3.23. Let r, s ∈ N, γ ∈ Nm, r − |γ| = s. Let ∥ · ∥Y be a norm

on P(Rm), and let ∥ · ∥Yr be the norm defined in (3.16), and let Yr be the

completion of (P(Rm), ∥ · ∥Yr). Then there exists one and only one linear

and continuous operator of Yr to Ys which coincides with the ordinary partial

derivation of multi index γ on the elements of P(Rm). By abuse of notation,

we shall denote such operator by Dγ, just as the usual partial derivative of

multi index γ. We have

Dγy = limj→∞

Dγpj in Ys, whenever limj→∞

pj = y in Yr, (3.17)

and

∥y∥Yr =∑

|γ|≤r, γ∈Nm

∥Dγp∥Y , ∀ y ∈ Yr.

With analogy with the usual derivations, Dy denotes the matrix

(D1y, . . . , Dmy).

Then we state the following result of Lanza de Cristoforis [30, Thm. 4.1].

Theorem 3.24. Let r ∈ N\0. Let ∥ · ∥Y be a norm on P(Rm). Let Yr be

the completion of P(Rm) with respect to the norm ∥ · ∥Yr defined in (3.16). Let

X be a real commutative Banach algebra with unity. Let X be a real Banach

space. Assume that there exists a linear continuous and injective map J of X

into X . Let (·) ∗ (·) be a continuous and bilinear map of X × X to X . Let ‘∗’

satisfy the following condition:

J [x1] ∗ x2 = J [x1, x2], ∀x1, x2 ∈ X . (3.18)

Let A be an open subset of Xm. Assume that there exists an increasing function

ψ of [0,+∞) to itself satisfying condition (3.13), for all (p, x) ∈ P(Rm)×A.

Then the restriction of the map A of Theorem 3.17 to Yr × A is of class Cr

from Yr ×A to X . (Note that Yr ⊆ Y0, and that Y0 equals the space Y defined

74


in Theorem 3.17.) Furthermore, the differential of A at each (y#, x#) ∈ Yr×A

is given by

(u, v) 7−→ A[v, x#] +m∑l=1

A[Dly#, x#] ∗ wl,

for all (u, v) ≡ (u, (w1, . . . , wm)) ∈ Yr × Xm. (For the definition of Dly#, see

Theorem 3.23)

Now that we have introduced the above result on the r times differentiabil-

ity of A, we introduce a formula for the differentials dsA of order s = 1, . . . , r

of A of [30, p. 932]

Proposition 3.25. Let all the assumptions of Theorem 3.24 hold. Let r, s ∈ N,

1 ≤ s ≤ r. The differential of order s of A at (y#, x#) ∈ Yr × A, which can

be identified with an element of L(s)(Yr ×Xm, X ), is delivered by the formula

dsA[y#, x#]((v[1], w[1]), . . . (v[s], w[s])) =

=s∑

j=1

m∑l1,...,lj ,...,ls=1

A[Dls · · · Dlj · · ·Dl1v[j], x

#]∗(ws,ls · · · wj,lj · · ·w1,l1

)+

+m∑

l1,...,ls=1

A[Dls · · ·Dl1y

#, x#]∗ (ws,ls · · ·w1,l1) , (3.19)

for all (v[j], w[j] ≡ (wj,1, . . . , wj,n)) ∈ Yr × Xm, j = 1, . . . , s. In (3.19), the

symbols l1, . . . , ls denote summation indexes ranging from 1 to m.

Next we return to the applications to Sobolev Morrey spaces, and we prove

the following.

Proposition 3.26. Let r ∈ N\0. Let Ω1 be a nonempty bounded interval of

R. Then Cr+1(Ω1) is a completion of the space (P(R, ∥ · ∥Yr), where

∥p∥Yr ≡r∑

l=0

∥∥∥∥ dldtl p∥∥∥∥C0,1(Ω1)

∀ p ∈ P(R).

If f ∈ Cr+1(Ω1) and if pjj∈N is a sequence of P(R) which converges to f in

the ∥ · ∥Yr-norm and if l ∈ 0, . . . , r, then

dl

dtlf = lim

j→∞

dl

dtlpj, (3.20)

in Cr−l+1(Ω1).

75


Proof. As we have already pointed out

∥f∥C0,1(Ω1)= ∥f∥C1(Ω1)

∀ f ∈ C1(Ω1).

Hence,

∥p∥Yr =r∑

l=0

(∥∥∥∥ dldtl p∥∥∥∥C0(Ω1)

+

∥∥∥∥ dl+1

dtl+1p

∥∥∥∥C0(Ω1)

)∀ p ∈ P(R),

and

∥p∥Cr+1(Ω1)≤ ∥p∥Yr ≤ 2∥p∥Cr+1(Ω1)

∀ p ∈ P(R).

Hence, the norm ∥ · ∥Yr is equivalent to the norm ∥ · ∥Cr+1(Ω1)on P(R).

Since Cr+1(Ω1) is a Banach space and the restriction map which takes

p in P(R) to p∣∣Ω1

in Cr+1(Ω1) is linear isometry of (P(R), ∥ · ∥Cr+1(Ω1))

onto(p∣∣Ω1

: p ∈ P(R), ∥ · ∥Cr+1(Ω1)

), it suffices to show that for each f ∈

Cr+1(Ω1), there exists a sequence of polynomials pjj∈N in P(R) such that

f = limj→∞

pj∣∣Ω1

in Cr+1(Ω1),

i.e., p∣∣Ω1

: p ∈ P(R) is dense in Cr+1(Ω1). We already know that such a

statement is true for r = 0. We now assume that the statement holds for r

and we prove it for r + 1.

By inductive assumption, there exists a sequence of polynomials qjj∈Nsuch that

limj→∞

qj∣∣Ω1= f ′ in Cr(Ω1).

Now let x0 ∈ Ω1. Then

f(x) = f(x0) +

x∫x0

f ′(t)dt ∀x ∈ Ω1.

Then we set

pj ≡ f(x0) +

x∫x0

qj(t)dt ∀x ∈ Ω1.

Clearly, pj ∈ P(R) for all j ∈ N. Since limj→∞

qj∣∣Ω1= f ′ uniformly in Ω1, the

inequality

|f(x)− pj(x)| ≤ m1(Ω1)∥f ′ − qj∥C0(Ω1)≤

≤ m1(Ω1)∥f ′ − qj∥Cr(Ω1)∀x ∈ Ω1,

76


shows that limj→∞

∥f ′ − pj∥C0(Ω1)= 0. Hence,

limj→∞

∥f ′ − pj∥C0(Ω1)+

r∑l=0

∥∥∥∥ dldtl f ′ − dl

dtlqj

∥∥∥∥C0(Ω1)

= 0

and accordingly

limj→∞

∥f ′ − pj∥Cr+1(Ω1)= 0

Equality (3.20) is a well-known corollary of the theorem on passing to the limit

under the differentiation sign.

Remark 3.27. Under the assumptions of the previous theorem, the oper-

ator Dγ defined by (3.17) coincides with the ordinary Dγ-differentiation in

Cr+1(Ω1).

Theorem 3.28. Let p ∈ [1,+∞], λ ∈[0, n

p

], (1+λ) > n


open subset of Rn which satisfies the cone property. Let Ω1 be a bounded open

interval of R. Then the composition operator T from Cr+1(Ω1)×W 1,λp (Ω,Ω1)

to W 1,λp (Ω) defined by

T [f, g] ≡ f g ∀ (f, g) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1)

is of class Cr. If (f0, g0) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1), then the first order differ-

ential of T at (f0, g0) is given by the formula

dT [f0, g0](v, w) = v g0 + f ′(g0)w

for all (v, w) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1).

If s ∈ 1, . . . , r, then the s-th order differential of T at (f0, g0) is given by

the formula

dsT [f0, g0][(v[1], w[1]), . . . , (v[s], w[s])] =

=s∑

j=1

ds−1v[j]dts−1

g0w[1] . . . w[j] . . . w[s] +dsf0dts

g0w[1] . . . w[s],

for all (v[1], w[1]), . . . , (v[s], w[s]) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1).

Proof. We set ∥ · ∥Yr = ∥ · ∥C0,1(Ω1), X = X = W 1,λp (Ω), A = W l,λ

p (Ω,Ω1), J

equal to the identity map, m = 1. As we have shown, Cr+1(Ω1) is a completion

of (P(R), ∥ · ∥Yr). By Corollary 3.16, W 1,λp (Ω) is a Banach algebra and there

77


exists a function ψ as in (3.10). As we have already proved in the proof of

Theorem 3.19, the abstract composition A of Theorem 3.17 coincides with T .

Then we can invoke Theorem 3.24 and Proposition 3.25 conclude that T is

of class Cr from Yr × A = Cr+1(Ω1) ×W 1,λp (Ω,Ω1) to W

1,λp (Ω) and that the

formulas for the differentials hold.

78

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