on discrete fractional integral operators and related diophantine equations

ON DISCRETE FRACTIONAL INTEGRAL OPERATORS

AND RELATED DIOPHANTINE EQUATIONS

JONGCHON KIM

Abstract. We study discrete versions of fractional integral operatorsalong curves and surfaces. lp lq estimates are obtained from upperbounds of the number of solutions of associated Diophantine systems.In particular, this relates the discrete fractional integral along the curve(m) = (m,m2, ,mk) to Vinogradovs mean value theorem. Sharplp lq estimates of the discrete fractional integral along the hyperbolicparaboloid in Z3 are also obtained except for endpoints.

1. Introduction

The classical Hardy-Littlewood-Sobolev inequality gives the Lp(Rd) Lq(Rd) bounds for the fractional integral operator f | |d f for 0 < < 1, 1 < p < q < , and 1q = 1p (1 ). The lp(Zd) lq(Zd) boundsfor its discrete analogue f mZd\0 |m|df( m) follows from theLp(Rd) Lq(Rd) bounds by a simple comparison argument [15]. However,if one considers discrete analogues of recent variants of fractional integrals,where the integration is taken over a sub-manifold, the argument fails andthe problem becomes more interesting.

Let us give a few examples. Consider the operators Ik, and Jk, definedby

Ik,(f)(n) =m=1

f(nmk)m

,

Jk,(f)(n1, , nk) =m=1

f(n1 m,n2 m2, , nk mk)m

.

The study of lp lq estimates of the operator Ik, and Jk, was initiated byStein and Wainger [15], where they obtained almost sharp bounds for k = 2and 1/2 < < 1 by employing the circle method. Oberlin [11] obtainedsharp results for k = 2 and 0 < < 1 except for endpoints without usingthe circle method. The endpoint bounds were fully established in a series ofpapers by Stein and Wainger [15, 16], and Ionescu and Wainger [9], but onlyin the case k = 2. The case k 3 seems to be substantially harder. Pierce[13] studied the operator Ik, for k 3 using tools from number theory, but

Supported in part by Study Abroad Scholarship Program by NIIED and NSF grant1201314.

1

2 JONGCHON KIM

it is open for k 3 in the full p, q range. See [14] for a generalization of J2,to quadratic forms in Zd.

In this paper, we shall consider the operator J acting on (initially) com-

pactly supported functions f : Zd C by

J (f)(n) =

m=1

f(n (m))m

,

where : N Zd is an injection. Note that Ik, and Jk, are special casesof J .

It is known [15, Proposition (b)] that the operator J extends to a

bounded operator from lp(Zd) to lq(Zd) if 1q 1p(1) and 1 < p < q 0, we have Nr (P ;h)P r+ for each > 0 uniformly in h Zd. Let s N be the number such

DISCRETE FRACTIONAL INTEGRALS 3

that we have either r = 2s or r = 2s 1. Then J extends to a boundedoperator from lp(Zd) to lq(Zd) if 1 r < < 1 and p, q satisfy

(i) 1q s (1 ), 1q < 1 s (1 ).

Several remarks are in order.

(1) When r = 2s, it is enough to assume that N2s(P ; 0) P 2s+ sinceN2s(P ;h) N2s(P ; 0) for all h Zd. This can be seen by writing

N2s(P ;h) =

[0,1]d

|S()|2se(h )d,

where e(t) e2piit and S() = Pm=1 e((m) ).(2) Given the stronger estimate Nr (P ;h) P r, it is possible to re-

place < in (i) by with a slight modification of the proof of Theorem1.1. This can be done by applying an abstract analogue ([2, Section6.2]) of an interpolation argument of Bourgain [1], but we shall notpursue it here.

(3) If one considers operators with summation over m Z \ 0, then thecondition xi [1, P ] Z in (1.2) changes to xi [P, P ] Z. SeeSection 4 for an analogous statement in higher dimensions.

Before we turn to the proof of Theorem 1.1, we shall give applications forthe case a(m) = (ma1 ,ma2 , ,mad) by using Vinogradovs mean valuetheorem in Section 2. We prove Theorem 1.1 and a sharp (up to endpoints)lp lq bound for the discrete fractional integral along the hyperbolic pa-raboloid in Z3 in Section 3 and Section 4, respectively. In Appendix 5.3, wegeneralize the Fourier multiplier approach [13] for operators Ja consideredin Section 2, giving an alternative proof of Theorem 2.1.

2. The operator Ja

We study the operator Ja Ja

, where a(m) = (ma1 , ,mad) for a

d-tuple of strictly increasing natural numbers a = (a1, , ad). We definea = a1 + + ad.Conjecture 1. Let 0 < < 1. Ja extends to a bounded operator from

lp(Zd) to lq(Zd) if and only if p and q satisfy(i) 1q 1p 1a(1 ) and(ii) 1p > 1 , 1q < .

See Appendix 5.1 for the necessity of conditions (i) and (ii). One is mainly

interested in proving Conjecture 1 for the range of a12a1 < < 1, sincethen one may get the full result by interpolating the result with the triviall1(Zd) l(Zd) bound for

4 JONGCHON KIM

In view of Theorem 1.1, the operator Ja is related to the quantityNar (P ;h).

For even r = 2s, let us denote Na

2s (P ; 0) by Jas(P ), i.e. the number of solu-

tions of the Diophantine system

xaj1 + + xajs = xajs+1 + + xaj2s

for 1 j d with xi [1, P ]Z. By a standard argument (see [6] or [18]),one may show that there is a lower bound

(2.1) P s + P 2sa Jas(P ).It is natural to ask if P s+P 2sa is the true order of Jas(P ). This question

for the case a = ak (1, 2, , k) has been of great interest. Non-trivialupper bounds for Js,k(P ) Jaks (P ) are collectively known as Vinogradovsmean value theorem. Recent work by Wooley [19, 20] and Ford and Wooley[7] report the following substantial progress on Vinogradovs mean valuetheorem.

Theorem A. Suppose that s and k 3 are natural numbers and that 1 s (k+1)24 or s k2 1. Then for each > 0, one has(2.2) Js,k(P ) P (P s + P 2s

k(k+1)2 ).

Thus, Theorem A answers the question up to if s is sufficiently larger

and smaller than ak = k(k+1)2 . Let V (k) be the least number s ak forwhich (2.2) holds. A crude standard estimate (see Appendix 5.2) gives

(2.3) Jas(P ) P 2sa+for s V (ad). However, we expect that (2.3) holds for a larger range of sin view of the lower bound (2.1). We denote by V (a) the least number s forwhich (2.3) holds. With r = 2s = 2V (a) and = a, Theorem 1.1 impliesthe following:

Theorem 2.1. Let ad 3 and a = 1 a2V (a) . Then Ja extends to abounded operator from lp(Zd) to lq(Zd) if a < < 1 and p, q satisfy

(i) 1q V (a)a (1 ), 1q < 1 V (a)a (1 ).

If one has V (a) = a, then Theorem 2.1 would imply the nearly sharpresult toward Conjecture 1 for 12 < < 1. In order to extend this to the fullrange of 0 < < 1, we need the stronger estimate

(2.4) Na2a1(P ;h) P a1+

uniformly in h Zd. Indeed, Oberlins result for J2, is based on the estimate(2.4) which is valid for a = (1, 2). See [11] for details.

We record the results for the special case Jk, = Jak , where ak = (1, 2, , k)

from Theorem A. Theorem 1.1 with r = 2s = 2(k21) and = k(k+1)2 gives


1/p

1/q

0 13

49

34

1

14

718

23

1

Figure 1. The known region of boundedness of J3, 23

Corollary 2.2. Let k 3 and k = 1 k4(k1) . Then Jk, extends to abounded operator from lp(Zk) to lq(Zk) if k < < 1 and p, q satisfy

(i) 1q 2(k1)k (1 ), 1q < 1 2(k1)k (1 ).

Theorem 1.1 with r = 2s = 2 = 2b (k+1)24 c impliesCorollary 2.3. Let k 3 and 12 < < 1. Then Jk, extends to a boundedoperator from lp(Zk) to lq(Zk) if p, q satisfy

(i) 1q 1 , 1q < .Corollary 2.3 complements Corollary 2.2 in the sense that it allows a wider

range of applicable and that the condition (ii) is optimal at the expenseof relaxing condition (i).

Figure 1 illustrates Corollary 2.2 and 2.3 for the case k = 3 and = 23 . It

shows the known (shaded) and the conjectured range of exponents (1p ,1q )

[0, 1]2 where J3, 23

is bounded from lp(Z3) to lq(Z3). Corollary 2.2 and 2.3give the vertices (49 ,

718) and (

13 ,

14), respectively.

2.1. Relation toWarings problem. The purpose of this section is twofold;to give a slight improvement to one of the results on the operator Ik, in[13] and to give a correction to an interpolation lemma in [13]. We thankPierce for encouraging us to clarify the issue here.

6 JONGCHON KIM

We denote by rs,k(l) the number of representations of l as a sum of spositive k-th powers. Then there is an estimate

(2.5) rs,k(l) ls/k1+for every sufficiently large s with respect to k, see [18]. Let G(k) to be theleast natural number s for which the estimate (2.5) holds. Since the work by

Hardy and Littlewood that G(k) (k2)2k1+5, numerous improvementshave been achieved. We refer the reader to [7] and references therein.

In [13], (2.5) was applied to obtain estimates on the mean values of theWeyl-sums. Instead, we use it to obtain estimates on N2s1(P ;h), which isthe number of the solutions of the Diophantine equation of 2s 1 variables(2.6) xk1 + + xks = xks+1 + + xk2s1 + hfor xi [1, P ] Z and h Z. Indeed, one has(2.7) N2s1(P ;h) P 2s1k+

for s G(k) uniformly in h Z.For the convenience of the reader, we record the argument here. One first

considers the number of solutions (x1, , xs) for each fixed (xs+1, , x2s1).It is O(P sk+) uniformly in h by (2.5) since we may assume that the righthand side of (2.6) is O(P k). We get (2.7) since there are P s1 many choicesfor (xs+1, , x2s1).

The estimate (2.7) and Theorem 1.1 with r = 2G(k)1 give the followingwhich slightly improves the allowable range of in [13].

Theorem 2.4. Let k = 1 k2G(k)1 . Then Ik, extends to a boundedoperator from lp(Z) to lq(Z) if k < < 1 and p, q satisfy

(i) 1q G(k)k (1 ), 1q < 1 G(k)k (1 ).

We note that the optimal condition (ii-c) 1/q < , 1/p > 1 in thestatement of [13, Theorem 4] should be replaced by the weaker condition(ii) in Theorem 2.4. This is due to an error in the interpolation lemma [13,Lemma 2]. The condition (ii) 1/q < , 1/p > 1 in the lemma should becorrected to a weaker condition (ii) 1/p > 12(1) , 1/q < 1 12(1) .

3. Proof of Theorem 1.1

The theorem can be reduced to certain restricted weak type estimates.We decompose the operator J dyadically. Define J

,j by

J,j(f)(n) = 2j

m

f(n (m))

for j 0 where m denotes 2jm


Lemma B (Christ [4]). Suppose that T is an operator taking characteristicfunctions E onto measurable functions with TE 0 for any measurable setE. Given > 0 and E with 0 < |E| 0.Let T = J,j , > 0, E Zd and F F j , j , Ek Ejk, Fk F jk be as

in Lemma B. We may assume that |F | > 0.Note that n Ek implies

(3.1)m

Fk1(n+ (m)) 2jk1

and n Fk implies(3.2)

m

Ek(n (m)) 2jk1

for k 1.When r = 2s 1, we define the sum Sr by

Sr =m1

m2

mr

E

(n+

ri=1

(1)i(mi))

for a fixed n Fs1.Since n Fs1, there are at least 2js2 values of m1 [2j , 2j+1) such

that n (m1) Es1 by (3.2). For each of these m1, there are at least2js2 values of m2 [2j , 2j+1) such that n (m1) + (m2) Fs2 by(3.1). Continuing in this manner, we get the following lower bound of Sr:

(3.3) Sr & 2(2s1)jss1 & 2rjr|F |s1|E|s1

since |F ||E| and r = 2s 1.Next, we get an upper bound for Sr. Let E

= n E.

Sr =m1

m2

mr

E

(ri=1

(1)i+1(mi))

=lE

r

i=1(1)i+1(mi)=l2jmi 0 so that = r+2r . (3.3) and (3.4) give

r|F |s1 2j |E|s,

8 JONGCHON KIM

or equivalently,

(3.5) r|{J,j(E)(n) > }|s1 2j |E|s.Since J (E)(n)

j=0 J

,j(E)(n), (3.5) implies

r|{J (E)(n) > }|s1 |E|s,which is equivalent to the restricted weak-type (2s1s ,

2s1s1 ) estimate for J

.

When r = 2s, we take the sum Sr by

Sr =m1

m2

mr

E

(n+

ri=1

(1)i+1(mi))

for a fixed n Es. After a few similar estimates, we get the restrictedweak-type ( 2ss+1 , 2) estimate for J

.

Moreover, the same estimates are valid for a complex-valued as long as 1 r . For each fixed such , we first obtain strong type estimateswith bounds uniform in =() by real interpolation with the l1 l boundfor 1 , 1q < .This result is sharp up to endpoint. One can show the necessity of the

condition 1q 1p 12(1 ) by taking f(n) = |(n1, n2)||n3| for nj 1and f(n) = 0 otherwise, for some appropriate , > 0. The necessity ofcondition (ii) can be shown as in Appendix 5.1.


To treat the operator P, we consider a variant of J defined by

J (f)(n) =

mZd0\0

f(n (m))|m|d0

acting on (initially) compactly supported functions f : Zd C, where : Zd0 Zd is an injection.

For fixed h Zd, r, P N, let N r (P ;h) denote be the number of solutionsof the Diophantine system

ri=1

(1)i+1(mi) = h

for mi BP , where BP = {x Zd0 : |x| P} and |x|2 = x21 + + x2d0 .We have the following variant of Theorem 1.1.

Theorem 4.2. Suppose that for each > 0 we have N r (P ;h) P d0(r)+for a fixed r N and > 0 uniformly in h Zd. Let s N be the numbersuch that we have either r = 2s or r = 2s1. Then J extends to a boundedoperator from lp(Zd) to lq(Zd) if 1 r < < 1 and p, q satisfy

(i) 1q s (1 ), 1q < 1 s (1 ).

The proof of Theorem 4.2 is a straightforward modification of the proofof Theorem 1.1 and will be omitted.

Proof of Theorem 4.1. By Theorem 4.2, it is enough to show that

(4.1) N 3 (P ;h) P 2+

uniformly in h Z3.Here, d0 = 2, m = (m1,m2) Z2 and (m) = (m, (m)), where (m) =

m21m22. Recall that N 3 (P ;h) is the number of solutions of the Diophantinesystem

x+ y = z + v

(x) + (y) = (z) + t

for x, y, z BP and h = (v, t) Z2+1.Since there are O(P 2) many z in BP , it is enough to show that the number

of solutions of

x+ y = v

(x) + (y) = t(4.2)

for x, y BP is O(P ) uniformly in h = (v, t) Z2+1.Suppose that (x, y) is a solution of (4.2). Then |v| 2P and |t| 2P 2.

We make a change of variables X = 2x v and Y = 2y v. Then (X,Y )

10 JONGCHON KIM

is a solution of the Diophantine system

X + Y = 0

(v +X) + (v + Y ) = 4t,

which is equivalent to the Diophantine equation

(4.3) (X1 +X2)(X1 X2) = N := 2t (v12 v22),

where X = (X1, X2) and v = (v1, v2).

Therefore, the number of solutions (X1, X2) of (4.3) is O(d(N)), whered(N) is the number of divisors of N . The fact that d(N) = O(|N |) (seeChapter 18 of [8]) and |N | = O(P 2) implies that the number of solutions(X1, X2) of (4.3) is O(P

) for any > 0, which in turn implies that thenumber of solutions of (4.2) is O(P ).

Theorem 4.2 with r = 3 and = 2 implies Theorem 4.1 for 13 < < 1.

Interpolating the result with the trivial l1(Z3) l(Z3) bound for 1 .For the necessity of the first condition, we take f : Zd C by

f(n) =

{ dj=1 |nj | if nj 6= 0 for all 1 j d

0 otherwise

for a fixed constant > 1/p so that f lp(Zd).For n1 > 1 and nj naj/a11 for 2 j d,

Ja(f)(n)

1ma1


Thus,

Ja(f)qlq(Zd) &n1>1

nq(1)/a1q1

naj1 n

a1j 2n

aj1

2jd

dj=2

nqj

&n1>1

nq(1)/a1q1

dj=2

naj(1q)/a11

&n1>1

nq(1)/a1+a(1q)/a111 ,

where the last sum converges only if 1q < 1a(1 ). We get the firstnecessary condition since we may decrease to 1/p as close as we want.

5.2. Proof of (2.3). Let a = (a1, , ad) be given. Let l = ad d and{bi}li=1 be the increasing sequence of natural numbers such that {b1, , bl} =([1, ad]Z)\{a1, , ad}. Considering the underlying Diophantine systems,we have

Jas(P ) =

|hb1 |sP b1

|hbl |sP bl

[0,1]ad

|S()|2sl

i=1

e(hbibi)d

Pl

i=1 biJs,ad(P ) = Pad(ad+1)

2aJs,ad(P ),

(5.1)

where S() =P

m=1 e(1m+ 2m2 + + admad).

Combining (2.2) and (5.1), we have

(5.2) Jas(P ) P 2sa+

for s V (ad).

5.3. Fourier multiplier approach for Ja . Let f() =

nZd f(n)e(n ) be the Fourier transform of f l1(Zd), where e(t) e2piit. We shallstudy the Fourier multiplier ma of the operator J

a

ma() =

n=1

e(a(n) )n

for [0, 1]d, given by the relation Ja(f)() = ma()f().By closely following the argument in [15, 13], we generalize the Weyl sum

approach on the Fourier multipliers (Proposition 5 of [13]) as follows:

Proposition 5.1. Let s N, 0 < 2s, and C. Jas(P ) P 2s+ foreach > 0 if and only if ma L2s([0, 1]d) for all 1 2s .

The folk lemma (Lemma 2 of [15]) implies the following result.

12 JONGCHON KIM

Corollary 5.2. Suppose that one has Jas(P ) P 2s+ for some 0 < 2s. Then the operator Ja extends to a bounded operator from l

2ss+1 (Zd) to

l2(Zd) for 1 2s .Note that Theorem 2.1 follows from Corollary 5.2 and a complex inter-

polation.

Proof of Proposition 5.1. For l Zd, let ras (l) be the number of solutions ofthe Diophantine system

(5.3) xaj1 + + xajs = lj ,

where l = (l, ld) = (l1, , ld) and xi 1 for 1 j d. In addition, wedefine ras (l;P ) to be the number of solutions of (5.3) for 1 xi P .

Let > 0 be given. We observe that Jas(P ) P 2s+ is equivalent to

(5.4)P

ld=1

Ras(ld) P (2s+)/ad , where Ras(ld) =

lZd1

(ras (l

, ld))2.

This follows from Parsevals identity applied to Jas(P ) =[0,1]d |S

a()|2sd,

where Sa() =

Pm=1 e(

a(m) ). Indeed, one hasJas(P ) =

lZd

(ras (l;P ))2 =

1ldsPad

lZd1

(ras (l, ld))2 =

1ldsPad

Ras(ld).

Therefore, it is enough to show that (5.4) is equivalent to

(5.5) ma L2s([0, 1]d), for every 1

2s.

As in [15, 13], one has

ma() = Ca,

10Say ()y

/addy

y+O(1),

where Ca, = (/ad) and Say () =

n1 e

nadye(a(n) ) which iswell-defined for each y > 0. This follows from the observation

0en

adyy/addy

y= n(/ad).

Thus,

(5.6) maL2s([0,1]d) Ca, 10

SayL2s([0,1]d)y


For the converse, it is enough to assume (5.5) only for R. Thenma()

s =

lZd ale(l ) L2, where

al =

n1, ,ns1

a(n1)++a(ns)=l

1

n1 ns.

Parsevals identity implies that

lZd |al|2 is finite. Since nadi ld in thesum, we have al ras (l)ls/add if ld 1. Thus,

>lZd|al|2

ld1

lZd1

(ras (l))2l2s/add =

ld1

Ras(ld)l2s/add

for every > 1 2s , which is equivalent to (5.4). Acknowledgments

The author is grateful to a referee for the valuable comments, whichimproved the results of the paper. The author would like to thank Jong-Guk Bak for suggesting this research topic and his guidance, and LillianPierce for her encouragement. The author also would like to thank JunghoPark for helpful discussion on the circle method and Stephen Wainger forpointing out a misquotation. This paper grew out of the authors Mastersthesis [10], and Theorem 4.1 is contained in the thesis.

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Department of Mathematics, University of Wisconsin-Madison, Madison,WI 53706 USA

E-mail address: [email protected]

on discrete fractional integral operators and related diophantine equations

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