representations of solutions of certain partial differential equations related to liouville's...

Representations ol Solutions of Certain Part ial Differential Equat ions

Related to Liouville's Equation

By ERWI~ KREYSZm, Windsor*)

I-rerrn Professor E. SP~RN~ zum 70. Geburtstag gewidmet

1. Introduction

We shall consider partial differential equations of the form

(1.1a) + c(z, z*)u = 0] where z, z* are independent complex variables and c is a general solution of Liouvilie's equation v,~, ---- e" times a constant; hence c is of the form

~' (z) ~' (z*) (1.1b) c(z, z*) = ~ p(z,z*)' ' p(z' z*) ---- ~0(z) + vd(z*),

where primes denote derivatives with respect to the corresponding independent variables and we impose on ~o and y the following condition.

(A) The c o ~ l e x /unctions q~ and ~p are arbitrary but such that c in (1.1)

is a holomorphic /unction o / z , z* in a domain ~2, (0, O)e •, in complex zz*-space, and is not identically zero.

Equation (1.1) and special cases of it were recently considered in a number of papers, in particular in connection with linear operators mapping complex analytic functions into solutions ~ of (1.1); we mention [1--4, 7--9, 11, 12, 15--18] in the list of references at the end of this paper.

Those investigations include attempts to determine explicit representations of kernels of B ~ R O ~ integral operators suitable for representing solutions of (1.1). Such representations are needed in connection with the coefficient problem, the study of the type and location of singularities, the growth of classes of solutions and other applications of B~RO~A_~T'S theory. The representations are also necessary in the

*) Work supported by the National Research Council of Canada, Grant Nr. A 9097.

Representations of Solutions of Certain Partial Differential Equations 33

transition to differential operators for developing a function theory of solutions of a given linear partial differential equation along the lines of complex analysis.

We shall consider the problem of obtaining explicit forms of B~RGMA~ kernels for equations of the form (1.1) with general ~0 and v 2 satisfying condition (A). We shall proceed as follows.

The next section contains concepts needed in connection with BV.R(~MAI~ operators for (1.1). In Sec. 3 we show that the transformation

= ~(z) , ~* = ~(z*)

which is suggested by the form of o in (1.1b), is of no help to our present problem. Then we define a class of partial differential equations and BERGMA~ operators (class P) which will be necessary in the transition from integral to differential operators. In Sec. 5 we present a solution of our problem, and in Sec. 6 we give another solution of the problem for equations (1.1) of class P. Sec. 7 contains the corresponding proof. Sec. 8 is devoted to differential operators. In the last section we extend the previous method to a more general class of equations and include a study of kernels of various degrees for the same class of equations.

2. Bergman operators

BERGMAN operators are linear operators defined on the complex vector space V(~l) of all functions / e C~(~1), where I2 x is a domain in the complex plane such that 0 e f2 x. Let Lu = 0 be a given homo- geneous linear partial differential equation of second order in two independent variables with coefficients atk, bj e C ~ (Q), where ~ --~ ~x • ~ and Q2 is ~x, regarded as a domain in the z*-plane. Then there exists a BV.RGMAN operator T: V(f2x) ---> S a ( L ) , where ~qn(L) is the vector space of all C~-solutions of L u -~ 0 on ~.

A theory of these and related operators has been developed by S. BE~GMAN [5] and others. A main purpose of such operators is a charac- terization of general properties of classes of solutions u by methods and results of complex analysis, for instance, with respect to the domain of regularity, behavior near singularities, distribution of values, growth on certain subsets of the space of independent variables, and the coefficient problem for various series representations; cf. the bibliography in [5]. These operators also have useful applications in hydrodynamics, for instance in connection with the Tricomi equation; cf. M.Z.v . KRZYWO- BLOCXX [14].

3 Hbg. Math. Abh., Bd. XLIV

34 Erwin Kreyszig

Equation (1.1) is of the form

(2.1) Lu -= u~. A- b(z, z*)u~. -4- c(z, z*)u --= 0,

where b, c e C'(/2), /2 ----/21 • 0 e/21- For this equation, a Berg- man operator T o can be defined by

1 (2.2) (Tg/)(z, z*) = f g(z, z*, t)/(�89189 dr;

- -1

here, ~ ---- 1 - - t ~ and t is real. g is called the kernel (or generating function) of T~. I f g is such that Tg: V(/2~)--->S~(L) with L given b y (2.1), we call g a Bergman kernel ]or (2.1).

Lemma 2.1. (S. B~ .R(~N [5]). Let g be a solution of

(2.3) Tg~.~- t-lg~. ~- 2ztLg -~ 0

on/2 X I, I = ( - -1 , 1),/2 ----/21X/22,

such that

(2.4) ~�89 --~ 0 (t -+ ~- 1)

uni/ormly in a neighborhood N o/ the origin in zz*-space and g , . / z t eC~ • I). Then

T , / ~ S~(L) ( / e C'(/2~))

where L is given by (2.1).

For later use we note that for (1.1), equation (2.3) takes the form

~' (z) ~' (z*) A = 0. (2.5) Mg = ~ g ~ . , - t-~g,. + 2zt g~. A- ~--p~,~.~ y]

3. A transformation of (1.1)

Our aim is explicit representations of Bergman kernels for (1.1). For this purpose the obvious transformation

(3.1) ~ = ~(z), ~* = ~(z*)

is of no help, as we shall explain. In terms of ~, ~*, equation (1.1) becomes simply

(3.2) @~c. -f (~ + ~ . ) ~ ---- 0.

I t is easy to obtain a Bergman kernel ~ for (3.2). In fact, Theorem 1 in [12] gives a ~ if~t = - - n ( n + 1 ) , n e N, as a polynomial in t with coefficients depending on z, z*, and for other X one obtains a power series


in t, with coefficients of a similar form. However, if (3.1) is applied to ~, one does not in general obtain a Bergman kernel for (1.1). This follows from

Lemma 8.1. I / g is a Bergman kernel/or an equation o[ the [orm

(3.3) L w = w~.o + c(z, z*)w = O,

c e C ~ (Q1 • Q2), and a trans]ormation

(3.4) zl =h~(z), z~=h~(z*)

is applied, then g becomes a Bergman kernel/or the trans]ormed equation i] and only i] hi(z) = yz, where y is a constant.

The proof is immediate.

The Riemann function of (1.1) can readily be obtained from that for (3.2) by means of (3.1), but a little reflection shows that this would not be of help to our problem either since in the transition from the Riemann integral representation of solutions to their Bergman representation, one would merely obtain a complicated integral representation of a Bergman kernel for (1.1), instead of an explicit representation.

4. Operators of class P

Earlier at tempts to solve the present problem were based on a class of Bergman integral operators which have the property that they can b e cast into a form free of integrals so that they become differential operators. These operators were introduced in [10S and investigated in [7, 8, 11, 12, 17, 18]. The operators and related concepts may be defined as follows.

An operator L of the form (2.1) is said to be ofdass P, written L e P, if there is a Bergman kernel g for (2.1) which can be represented in the form

(4.1) g(z, z*, t) = ~ q2~,(z, z*)t2~ ', (q2• =~ O, n e N). Iz~O

This g and the corresponding Bergman operator To are said to be of class P, written g e k(P) and Tg e B(P), respectively.

Note that (4.1) does not contain odd powers of t; this is no loss of generality, as can be seen from (2.2).

For this class of integral operators, the transition to differential operators is furnished by the following theorem (cf. Theorem 1 in [8]).

3"

36 ErwinKreyszig

Theorem 4.1. I / L e P, representation (2.2) with g given by (4.1) can be represented in the/arm

(T/)(z, z*) = ~ (2~)!q~(~, ~*)]c~-.,(z) ~,ffi0 2 ~ P !z2

(4.2)

where

(4.3) ](z) = ~ :'-" .=o~('~ + O! B(�89 ~ + �89 z"%

B is the beta ]unction and the a~'s are the coe/-fieients o/the Maclaurin series o / ] in (2.2).

In [8] this theorem was used to prove that a differential operator introduced by K.W. BAUV.R [1] for representing solutions of a special equation (1.1), namely,

en(n + 1) (4.4) uz,, + (1 + ezz*) 2 u = 0 (e = --~ 1)

is a Bergman operator of class P. On the basis of results by K.W. BAU~.R [2, 3] about (4.4) and the above Theorem 4.1, H. F L O R ~ and G. J x ~ [7] discussed the form of Bergman kernels for equations (1.1) of class P. They departed from a representation

n ~ , x �9 , /c , (z ,z )\

(4.5) g(~, ~*, 0 = Z ~ Z ~ / ~ ) " /~=0 x = 0

This was probably suggested by (4.4) since in that case, one simply has

2c 1 + ~zz*

and arrives at the explicit solution given in [8]. From (4.5) and (2.5) one obtains a reeursive system for the o~v~'s, for which an explicit solution has been given only in the case (4.4), and solving the system for more complicated equations (1.1) may perhaps be quite difficult.

5. A Bergman kernel for (1.1)

We now drop the restriction L ~ P and return to the general case of {1.1) satisfying assumption (A) in Sec. 1. We solve the problem posed in Sec. 1 by obtaining a representation of a Bergman kernel g for (1.1) in which all occurring functions are given explicitly, and only some constants remain to be determined by recursion in a particular case.

TheoremS.1. A Bergman kernel g for (1.1) with q~,~ satis]ying assumption (A) in Sec. 1 is

(5.1) g(z, z*, t) = 1 + ~ z~(z) l~(z , z*)t~ p=l


where

(5.2)

and

z,~(z) = ( - - 2 z ) , / 1 . 3 . . . (2/~ - - 1),

(a) t~(z, z*) = Z p(z, z*)-~h~(9(z)) , ~ = 1, 2 , . . . (5.3) ,=i

(b) h~(9(~)) = ~ 2.~,~ ~ CV(z) , ~ = ~ " " "' ~' Ivl=~ i=1 p = 1, 2 , . . . ;

here, 7 = (71 , " " , 7v) , n ,I = 71 + . . . + 7, , c v , , = dv~/dzv~, a n d the 2~,~, y s are constants .

P r o o f . Substituting (5.2) into (5.1) and the resulting representation into (2.5), we obtain

10 = 1, l ~ , ~ . = 12~-2,~. + A9 ~ P t~_~ ~- 0 .

Integration with respect to z* gives

(5.4) l~, = l~i,_2,, + 2 9 ' f 12~_~p-~dp, I~ = 1, 2, . . . .

We now proceed by induction. Taking/~ = 1, we have

/~_~ ~' 1~ = 4 9 ' d p = __ 2 _~.

This is (5.3), where h11(9(z ) ) = - - 2 9 ' (z), hence 4111 = - - 4 . Suppose that (5.3) holds for some p. Then from (5.4), with/~ + 1 instead of/~, we have

12V+Z = 12a, z + 49' f 12~,P-2dP

Z t -v (h.~p - - ~hLp -~-1)

t - v ~9 ! = h~,~p - - ~ a - - 1 - - hl, ' ~_ip-o

~+I

= Z h ~ §

where primes denote derivatives with respect to z, and

with the convention hg~ ---- 0 when z ~ 0 or z > / ~ on the right. From this and (5.3b), applied to h~ and h~,~_ 1 on the right, we see that h~§ can be written in the form (5.3b), that is, there are eonstanCs 2~+1,~, r such that

h#+l, ' (9(Z)) = Z 2#+1, v, 7 I~I 9(?i)(Z) �9 J~l=~,+1 i=I

These constants can be determined recursively. This completes the proof.

38 Erwin Kreyszig

6. Equations (1.1) of class P

In general, Theorem 5.1 will not yield a Bergman kernel of class P for (1.1) with L e P. Nevertheless, the idea of proof of Theorem 5.1 will be helpful for obtaining kernels g e k(P) when L e P in (1.1).

I f ~t = - - n (n + 1) with n e N, then L e P in (1.1).

This was proved by W. WATZLAWEK [17]. A second proof follows by applying our Theorem 4.1 to a result (Theorem 3) by K.W. BAUER and G. JACK [4]. In fact, that Theorem 3 contains a differential operator for representing solutions of (1.1) with ~t ----- - - n(n ~ 1), n e N, and Theorem 4.1 implies that such an operator can be transformed into a Bergman operator of class P.

Neither of these two proofs leads to an explicit representation of a corresponding Bergman kernel g e k (P). In [4] the difficulty is that the derivatives of the function to which the operator is applied do not occur in separate terms. The following theorem removes this deficiency; it gives a g = go e k(P) in explicit form. go looks slightly more complicated than g in Theorem 5.1. This is due to the occurrence of functions ~n,~ to be determined from a system of equations [(6.4), below]; the solution of the system is obtained immediately by solving one equation after the other, without encountering simultaneous equations.

Theorem 6.1. In (1.1)let (A) be satisfied and

(6.i) ~ = - - n ( n + 1), n e N.

Then there exists a corresponding Bergman kernel go e k(P), namely,

n

(6.2) go(z,z*,t) -~ 1 Jr ~ Z21,(Z)~2~,(Z,Z*)t~I" p f f i l

where )~l, is given by (5.2) and

~ 0 ~ 0 x ' ' l

here the l~_2~'s are given by (5.3), and the ~ , / s are the unique solutions of

n §

A,h,~ = 0 , i = 0 , 1 , . . . , n (6.4) s~j

r ~ O

where boo -~ 1, s ~- 1, and hsj is given by (5.3b). Henc~e i/ (6.1) holds, then L e P in (1.1).

l~epresentations of Solutions of Certain Partial Differential Equations 39

7. Proof of Theorem 6.1

Let (6.1) hold for a fixed n e N. Theorem 5.1 will not give a kernel of class P , in general. However we may use the idea of its proof in a suitably modified form. That proof was based upon an operator J defined by

(7.1) J q = q, -4- , ~ ' f P-~qdv2

where q is a function of z, z*. In Theorem 5.1 we have

(7.2) [2~+2 = Jl2~, la = O, 1, ...

with the understanding that l 0 ---- 1 and the function of integration in (7.1), which is an arbitrary function of z, is chosen to be zero. For a function h depending only on z we have

(7.3) J h = h' ~ 12h.

Hence

(7.4) J(hq) -= h' q + h J q .

Another auxiliary formula of importance is

(7.5) J(p-~) ----- - - (v + v--~l) ~ ' , -~-1 ,

Since ~' =4= 0, it follows that

(7.6) J (p - ' ) = 0 ~ 2 - - - - - -v(v + 1).

v = 1, 2 , . . . ,

The sequence of functions (l~) in Theorem 5.1 obtained from (7.1) by choosing all functions of integration to be zero will, in general, not termi- nate, even i f L e P. However, if (6.1) holds for a fixed n e •, we can aug- ment t h e / ~ ' s in Theorem 5.1 by suitable functions of integration so that we get a polynomial of degree 2n in t as a Bergman kernel for (1.1). In fact, we show that there is such a kernel of the form (5.1) with l~,(z, z*) replaced by Z2~(z, z*) such that

(7 .7 ) = o > n ) .

From (7.1) and (7.2) we see that it suffices to prove

(7 .8 ) = 0 .

To obtain ~,+2 satisfying (7.8), we introduce at each step of the previous recursion a suitable function of integration, that is, instead of (7.2) we now take

(7.9) ~2~+2 (z, z*) ----- ~,, .+1 (z) + (JZ2.)(z, z*), p ---- 0, 1, . . .

40 Erwin Kreyszig

with ~n,1, ~bn,2,-.. tO be determined in a suitable fashion. We show that the various powers of p and their coefficient functions in ~.+~ are such that we can choose functions ~ ,1 , It,,2, ... of z in (7.9) such tha t (7.8) holds. The highest negative power o f p in ~2.+2 is p-~-l. From (5.3) we see tha t it is multiplied by

where

hn+l , n+l ~ ~n+l , ~t+l; 1 . . . . . 1~ 0tn+l

~n+1 ,n+ l ;1 . . . . . 1 = ( - - 1 ) n+l I - I (T + ~ , a ~ 0

and (7.5) shows that this is zero. We now turn to p-" in ~.+~. The function ~., 1 enters when ~2 is obtained and produces in Z2"+~ the function J ' ~ , 1. The latter has precisely one term involving p - ' , with coefficient

(--1)nq)'n~n,1/n!.

Since the function ~ , 1 is still arbitrary and the coefficients of the powers of p in ~.+~ depend only on z, we can choose s 1 such that the sum of the coefficients of all terms of ~2~+,. involving p-~ becomes zero. We now take ~ , 3, which is introduced in ~4 and produces in ~2,+~. the function J~-lit~,2. The latter has precisely one term involving p-,+l, with coefficient

- - 1 ) ! .

We can now choose Its, 2 so that the sum of the coefficients of all the terms of ~+~ involving 1o -~+1 becomes zero. In the next step we choose ft.,3 so that the sum of the coefficients of the terms containing p-'+~ becomes zero, etc. In the (n - - 1)th step we take care of 10-1 by choosing a suitable function Its.._1 and in the last step we choose/t,, ~ so that the sum of all the remaining terms (which depend only on z) becomes zero; this sum is

n , . - 1 "~- n ,n -2 "~ " '" "J[-..n,1 ,

as can be seen from (7.3). We finally have to prove that the ~., j's are obtained from (6.4). From

(7.3) we have by induction

(7.10) J v h - - ~ ' ["~ h(")l

with l~ given by (5.3a). Setting s --~ 1, we may write

/z

(7.11) ~2~ = ~ J'hn,~,_~. v ~ 0


From (7.10), (7.11), and (5.3a) we obtain (6.3). Setting ;u = n + 1, we have

v - - x

v ~ O x ~ O a ~ l

By collecting like powers of ID and equating to zero the sum of the coefficients of ID -~ (which are functions of z but do not depend on z*) we obtain (6.4). From (5.3b) in Theorem 5.1 we see that

and hj,~ # 0 for i = 1, . . . . n, so that there is no problem in solving tha t system. The solution is of the form

~ t , 1 : - - h ; 1 . hn+l,

s - - - - n s - - h . -1 , ._~(s ~h . , . _ l -b h.+1,.-1)

etc. This completes the proof of Theorem 6.1.

8. DiiTerential operators for (1.1)

A differential operator for (1.1) was given by K.W. BAUER [3]. That operator involves a complicated recursion. A simpler operator was obtained by K.W. BAUE~ and G. JANE [4]. I t is applied to a holomorphic function ] and is such that each derivative ]', ]" , . . . appears in various terms (and combined with positive and negative powers and derivatives of ~' of various orders). A differential operator for (1.1) in which the derivatives of ] appear in their natural order is given in the following theorem.

Theorem 8.1. A di~erential operator T /or representing 8o~u$ions u o~ (1.1) with 2 = - - n ( n ~ 1), n e N, is defined by

(s.1) u ( z , z * ) = = Z ~ 0

where ~o = 1 and ~l,, # = 1, . . . , n, is given by (6.3), and f is holomorphic in a neighborhood o/ the origin o/ the complex plane.

P r o o f . Since A • - - n(n ~ 1) where n e ~, Theorem 6.1 shows that L e P in (1.1) and there exists a corresponding Bergman kernel go e k(P), of degree 2n in t. By (6.2), that kernel is of the general form

g0(z, z*, t) = ~ q~(z, z*)t~. D = 0

From this and Theorem 4.1 the assertion follows.

42 Erwin Kreyszig

(9.3)

where

9. Relations between various Bergman kernels for (1.1) with L e P

By definition, if L e P, there is a Bergman kernel g for L u = 0 which is a polynomial in t. Given L, the g of smallest degree in t for L u = 0 is called a minimal kernel of class P /or that L. This notion was introduced by M. K r a e h t and G. S c h r S d e r [9], and its importance ea~ be seen from [9] and [18]. For similar ideas in connection with operators of exponential type, see also [13], p. 130.

The kernel in Theorem 6.1 is minimal for L, as follows from the proof. In Sees. 5 and 6 we have constructed different Bergman kernels for

the same equation of the form (1.1) with L e P. We want to show that the method in the proof of Theorem 6.1 can be used in a more general form for obta~fing various Bergman kernels g e k(P) of prescribed degree (greater than that of a minimal kernel) for a given equation

(9.1) L u = u~. ~- b(z, z*)u~. + e(z, z*)u = 0

with L e P ff a kernel g for (9.1) is known.

Theorem 9.1. Let L ~ P in (9.1) and let a corresponding Bergman kernel g ~ k (P) be given by

n

(9.2) g(z, z*, t) = lo(z ) Jr- ~ g~.~,(z)l,.j,(z, z*)t~'~ '

where g~l,(z) is defined by (5.2). Then another Bergman kernel ~ ~or (9.1) is given by

~(z, z*, t) = Io(Z) + ~ Z,.~,(z)~2t,(z, z*)t~ , u = l

Z~(z, z*) = ~, k~,(z)l~,(z, z*) (9.4) " = ~

here 12t, = 0 when l~ < 0 or # > n, and s h~, . . , are any junctions of z. For any given integer m > n there exist coeffivients Z~l , such that (9.3) reduces to a polynomial o/degree 2m in t.

P r o o f . The functions l~ in (9.2) satisfy

(9.5) 1~+2 ' ~. ---- L l ~ ,

where L is defined by (9.1). By integration,

(9.6) l~.+~ = Jl12~, = l~,,~ + fbl~., ~.dz* + fclz~,dz*.


This defines J1. A function of z remains arbitrary in each such integration. Hence we may obtain coefficients ~ for a Bergman kernel for (9.1) by setting

(9.7) ~o = / 0 , 1~+2 = J l l ~ , + Its,+1, tt = O, 1 . . . . ,

where s it2 . . . . do not depend on z*. We shall represent these new coefficients in terms of those in (9.2). From (9.6) and (9.7) we first have

/t

Z2~, --= ~ J~h~,_~, g = 1, 2 . . . . , v ~ 0

where ho = lo. The operator J~ has properties similar to those of J in See. 7. In particular, for any h depending only on z we have

(9.8) J i b = ~ (~) h{'Ols~_s,,, v -= 0,1 . . . . .

I t follows that

I , ~ 0 x~O ~''~

Collecting terms with like 12~, we obtain (9.4). We prove the second statement of the theorem. Let m > n be

given and s = m - - n e N. By assumption, lo, 12, . . . , l~ is a solution of the reeursive system (9.5) for (9.1) such that l ~ ~ 0 for all # > n. Hence if 8 ~ n, the functions

[12~ p = 0 , . . . , s - - 1

(9 .9) ~2~ = i12~ "4- 12,-~s # = s, . . . , n t l~_~, # -- n -{- 1, . . . , n A- s

constitute a solution of (9.5) with the desired propety ~ = 0 for all p > n + s = m. I f s > n, the idea of proof is similar.

We finally mention that another proof of the second statement of the theorem would follow from the analogue of Theorem 8.1 for (9.1). In fact, if we set

] = + . . . +

where a# is any function of z, we obtain a representation involving derivatives of f up to ft,+s}, and ff we transform the representation back into integral form by means of Theorem 4.1, the desired result follows.

R e f e r e n c e s

[1] K. W. BAUER, ~Tber eine der Differentialgleichung (1 -4- z~)*wxi -4- n(n + 1)w = 0 zugeordnete Funktionenthoorie. Bormer Math. Schr. Nr. 28. Math. Inst. Univ., Bona, 1965.

[2] K. W. BAUER, l~ber eine partielle Differentialgleichtmg 2. Orduung mit zwei uaabhfmgigen komplexen Variablen. Monatsh. Math. 70 (1966) 385---418.

44 Erwin Kreyszig, Representations of Solutions of Certain P a r t i a l . . .

[3] K. W. BA~rE~, ~ber Differentialgleiehungen der Form ~(z ,2 )wz~- -n (n -~ 1)w = 0. Monatsh. Math. 75 (1971) 1--13.

[4] K. W. B A v ~ und G. J~'cK, Differentialoperatoren bei einer inhomogenen elliptisehen Differentialgleiehung. Rend. Ist. Mat. Univ. Trieste 8 (1971) 1--29.

[5] S. B E ~ G ~ , Integral Operators in the Theory of Linear Partial Differential Equations. 2nd rev. print., Ergebn. Math. Grenzgeb. 28. Springer, Berlin 1969.

[6] S. BE~GM~, On an initial value problem in the theory of two-dimensional transonic flow patterns. Pacific J. Math. 82 (1970) 29--46.

[7] H. FT.ORI~_W und G. JA~r~, 1)olynomerzeugende bei einer Klasse yon Differen- tialgleichungen mit zwei unabhRngigen Vaxiablen. Monatsh. Math. 76 (1971) 31--37.

[8] M. KRACH~ und E. KR~YSZXO, Bergman-Operatoren mit Polynomen als Er- zeugenden. Mannseripta math. 1 (1969) 369--376.

[9] M. KRACH~r und G. SCHR()DER, Bergmansehe Polynom-Erzeugende erster Art. Manuscripta math. 9 (1973) 333--355.

[10] E. KR~u ~ber zwei Klassen Bergmanscher Operatoren. Math. Naehr. 87 (1968) 197--202.

[11] E. KP~YSZm, Bergman-Operatoren der Klasse P. Monatsh. Math. 74 (1970) 437--444.

[12] E. KR~YSZIG, On Bergman operators for partial differential equations in two variables. Pacific J. Math. 86 (1971) 201--208.

[13] E. KR~YSZIO, Zur Theorie der Exponentialoperatoren und der Bergman- Operatoren 1. Art. Bul. Inst. Polit. Iasi 15 (1972) 127--137.

[14] M. Z. v. K~WOBLOCX~, Bergman's Linear Integral Operator Method in the Theory of Compressible Fluid Flow. Springer, Wish, 1960.

[15] I. N. V~.xuA, New Methods for Solving Elliptic Equations. l~orth-HolIand Publ., Amsterdam, 1967.

[16] G. W~NECKE, Lrber die Darstellung yon LSsungen der partiellen Differential- gleichung (1 -}- 6z~)~w~i =- ~ - - se 2'~ Bonner Math. Sehr. Nr. 34. Math. Inst. Univ., Bonn 1968.

[17] W. WA~r~r.AW~K, ~-ber lineare par~ielle Differentialgleichungen zweiter Ord- hung mit Bergman-Operatoren der Klasse P. Monatsh. Math. 76 (1972) 356---369.

[18] W. WA~Z~W~K, Hyperbolische und paxabolisehe Differentialgleichnngen der Klasse P. Ges. Math. u. Datenveraxb. Bonn, Berieht 77, 1973, 147--179.

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