Solid Cauchy transform on weighted poly-Bergman spaces
SSOLID CAUCHY TRANSFORM ON WEIGHTEDPOLY-BERGMAN SPACES
R. EL HARTI„ A. ELKACHKOURI, A. GHANMI
Abstract.
The so-called weighted solid Cauchy transform, from inside the unitdisc into the complement of its closure, is considered and their basic propertiessuch as boundedness is studied for appropriate probability measures. The actionthe disc polynomials is explicitly computed and used to describe the range of itsrestriction on the weighted true poly-Bergman spaces.
The weighted Cauchy transform C µs f ( z ) := 1 π Z Ω f ( ξ ) z − ξ dµ ( z ); z / ∈ Ω , (1.1)for z in the complement of the closure of the unit disc Ω, Ω c := C \ Ω, associatedto given measure µ on a bounded Ω in the complex plane, can be seen as a specificadjoint of the classical weighted Cauchy transform on b Ω [3, p. 89]. The importanceof these operator lies in the fact that its kernel function is the fundamental solution ofthe ∂ operator and is closely connected to the Green function for Dirichlet Laplacianfor specific Ω.The description of the range of C µs , corresponding to specific µ , when acting on thedifferent standard spaces of analytic functions was investigated by many authors,see for instance [4, 11, 14, 15]. For the Bergman space A (Ω) of analytic functionson Ω, this problem has been solved by many authors, notably by Napalkov andYulmukhametov [14] for Jordan domains and by Merenkov [12, 13] for a large classof domains including those bounded by a Jordan curve ∂ Ω with area ( ∂ Ω) = 0 or theintegrable Jordan domains. Obviously, the restriction of the solid Cauchy transformto A (Ω) is injective and its image is contained in the space of analytic functionson Ω c with f ( ∞ ) = 0. Moreover, it is a continuous operator [12] from A (Ω) intothe special Bergman-Sobolev space B (Ω c ) defined as the space of all holomorphicfunctions in Ω c vanishing at infinity and whose derivative belongs to A (Ω c ), B (Ω c ) = { f ; f ∈ A (Ω c ); f ( ∞ ) = 0 } . More precisely, for domains with sufficiently smooth boundary, it is proved in [14]that the solid Cauchy transform C s := C s ; µ = dxdy being the area measure, mapsthe dual space A ∗ (Ω) of A (Ω) continuously onto B (Ω c ). In [12], it is proved that C s ( A ∗ (Ω)) = B (Ω c ) remains valid if Ω is a quasidisc, i.e., an interior of a Jordancurve J for which there exists M > diam ( J ( a, b )) ≤ M | a − b | for any a, b ∈ J . Here J ( a, b ) is an arc of the smaller diameters of J joining a and b . Forbounded Jordan domain, it turns out to be a sufficient and necessary condition [15]. Key words and phrases.
Weighted solid Cauchy transform; Disc polynomials; Weighted truePoly-Bergman spaces, Range. a r X i v : . [ m a t h . C V ] S e p R. EL HARTI, A. ELKACHKOURI, A. GHANMI
Namely, the Cauchy transformation is a surjective continuous operator from A ∗ (Ω)onto B (Ω c ) if and only if the domain Ω is a quasidisk. This characterization hasbeen next used in [8] to investigate the action of the Laplace transform on Bergmanspaces.In the present paper, we consider similar problem for the weighted solid Cauchytransform with respect to ω α ( | ξ | ) : (1 − | ξ | ) α , [3] C ω α s f ( z ) := 1 π Z D f ( ξ ) z − ξ ω α ( | ξ | ) dxdy ; z ∈ D c , (1.2)when acting on weighted true poly-Bergman spaces A ,γn ( D ) in the unit disc D ,which are specific generalization of the classical weighted Bergman space to thepolyanalytic setting. Thus, our main task is to determinate the range of C s whenacting on the n -th weighted true poly-Bergman space A ,γn ( D ) defined in [7] A ,γn ( D ) = H ,γn ( D ) (cid:9) H ,γn − ( D ) (1.3)for n ≥ A ,γ ( D ) = A ,γ ( D ). Here H ,γm ( D ) denotes the space of polyanalyticfunctions of order n + 1 that are square integrable on D with respect to given radialweight function ω γ ( | ζ | ) : (1 − | ζ | ) γ , H ,γm ( D ) := ker∂ n +1 z ∩ L ,γ ( D ), with ∂ z is thestandard Wirtinger derivative. The main result shows that the restriction of C sω α to A ,γn ( D ) is a finite dimensional vector space contained in the one spanned by z n − m +1 ; m = 0 , , , · · · , n. Its precise dimension depends on the quantization of γ − α andthe order of polyanalyticity n + 1. More precisely, it can be stated as follows Theorem 1.1.
Let γ > − and α > ( γ − / . Then, C sω α ( A ,γn ( D )) to A ,γn ( D ) isa finite dimensional vector of dimension N = dim( C sω α ( A ,γn ( D ))) = min( n, α − γ +1) + 1 when γ − α ∈ Z − and N = n + 1 . otherwise. Corollary 1.2.
Under the condition above, the null space of the restriction of C sω α to A ,γn ( D ) is spanned by the disc polynomials R γm,n ; m ≥ min( n, α − γ + 1) . Corollary 1.3.
The spaces C sω α ( A ,γn ( D )) n form an increasing sequence of spacesin L ,γ ( D ) . This note is outlined as follows. In Section 2, we briefly review from [17, 6] someneeded facts on the underlying true weighted poly-Bergman space A ,γn ( D ). Section3 is devoted to discuss the boundedness of the weighted solid Cauchy transform C µs for specific weight functions. In Section 4, we collect and establish needed toolson the action of C ω α s on the weighted poly-Bergman spaces A ,γn ( D ). The proof ofTheorem 1.1 and their corollaries is presented in Section 5. Let γ > − L ,γ ( D ) := L ( D , dµ γ ) the Hilbert space of complex-valued functions in the unit disk D = { z ∈ C ; | z | < } with the norm induced fromthe scalar product h f, g i γ := Z D f ( z ) g ( z )(1 − | z | ) γ dxdy ; z = x + iy ∈ D . Orthogonal decompositions of L ,γ ( D ) can be provided by means of the so-calledpolyanalytic functions [2] which are solutions of the generalized Cauchy–Riemann equation on the unit disc D , ∂ n +1 z f = ∂ n +1 f∂z n +1 = 12 ∂∂x + i ∂∂y ! n +1 f = 0 . In fact, we have the orthogonal Hilbertian decompositions L ,γ ( D ) = ∞ M n =0 A ,γn ( D ) and H ,γn ( D ) = n M k =0 A ,γk ( D ) , where the space A ,γn ( D ) are as in (1.3). They are closed subspaces of L ,γ ( D )andform an orthogonal sequence of reproducing kernel Hilbert spaces. An orthogonalbasis of A ,γn ( D ) is shown in [6] to be given by the disc polynomials [9, 10, 5, 1, 7] R γm,n ( z, ¯ z ) = m ! n ! m ∧ n X j =0 ( − j (1 − zz ) j j !( γ + 1) j z m − j ( m − j )! z n − j ( n − j )! (2.1)for varying m = 0 , , , · · · . Above, m ∧ n = min( m, n ) and ( a ) k = a ( a + 1) · · · ( a + k −
1) denotes the Pochhammer symbol. The completeness of the system R γm,n in L ,γ ( D ) was crucial in exploring the so-called weighted true poly-Bergman spaces.It should be mentioned here that for γ = 0, the spaces A , m ( D ) reduces further tothe poly-Bergman spaces studied in [16, 19, 18]. C s Let ω be a weight function on the segment (0 ,
1) such that the associated moment γ ωn := Z t n ω ( t ) dt ≤ γ ω ; n = 0 , , , · · · , (3.1)are finite. Associated to ω that we extend to a measure on the unit disc D inthe usual way by considering ω ( | z | ) for z ∈ D , we consider the correspondingweighted solid Cauchy transform [ C sω ( f )] ( z ) in (1.2) with Ω = D and z ∈ D c . Weconsider its action on the Hilbert space L ,A ( D ) := L ( D ; A ( | ξ | ) dλ ) of all complex-valued functions on D such that are square integrable with respect to given measure A ( | ζ | ) dλ , dλ being the Lebesgue measure. In a similar way, we define the Hilbertspace L ,B ( D c ) := L ( D c ; B ( | ξ | ) dλ ) for given weight function B ( | ξ | ) in D c , arisingfrom a weight function on (1 , + ∞ ). We denote by h· , ·i A and h· , ·i B the associatedscalar product and by k·k A and k·k B the associated associated norms, respectively.In the sequel, we provide sufficient conditions on ω , A and B to C s be a boundedoperator from L A ( D ) into L ,B ( D c ). Namely, we assume that V ω /A := Z ω ( t ) A ( t ) dt < ∞ (3.2)and W B := Z B (1 /t ) t (1 − t ) dt < ∞ . (3.3) Proposition 3.1.
Under (3.2) , the transform C sω is a well defined bounded operatorfrom L ,A ( D ) to Hilbert space L ,B ( D c ) . R. EL HARTI, A. ELKACHKOURI, A. GHANMI
Proof.
Using Cauchy Schwartz inequality, it is not hard to show that C sω is welldefined on L ,A ( D ). Indeed, for any f ∈ L ,A ( D ) we have |C sω ( f )( z ) | ≤ π Z D ω ( | w | ) | w − z | A ( | w | ) dλ ( w ) ! / k f k A ≤ (cid:16) V ω /A (cid:17) / π ( | z | − k f k A which is finite by means of (3.2) and the fact that | w − z | − , for varying w ∈ D ,satisfies | w − z | − ≤ ( | z | − − . Therefore, it follows kC sω ( f ) k B ≤ π V ω /A Z ∞ B ( r )( r − rdr ! k f k A ≤ π V ω /A Z B (1 /t ) t (1 − t ) dt ! k f k A . Then, one concludes for the boundedness of the solid Cauchy transform C sω from L ,A ( D ) into L ,B ( D c ) thanks to assumptions (3.2) and (3.3). (cid:3) Remark 3.2.
A precise estimate for the norm operator of the solid weighted Cauchytransform C sω can be given for explicit wight functions satisfying assumptions (3.2) and (3.3) . For example for A ( t ) = ω ( t ) = ω γ ( t ) = (1 − t ) γ and B ( t ) = B a,b ( t ) := t a ( t − b with − a < b < − < γ , the evaluation of the integrals giving V ω /A and W B yields kC sω k ≤ − b Γ(2 a + 2 b )Γ( − b − π ( γ + 1)Γ(2 a + b − . (3.4) Remark 3.3.
For A ( t ) = ω γ ( t ) and ω ( t ) = ω α ( t ) , the corresponding quantity V ω α /ω γ is finite if and only if γ < α . C sω α In order to explore the basic properties of C sω α such as the description of its nullspace and the range of its restriction to the true weighted polyBergman spaces, wespecify the weight functions A ( t ) = ω γ ( t ) = (1 − t ) γ and B , and we provide theexplicit action of C sω α on the disc polynomials R γm,n . The advantage of consideringsuch class of functions is that they form an an orthogonal basis of the Hilbert space L ,γ ( D ). To this end, we begin by giving its action on the generic functions e ‘jk ( ξ, ξ ) = ξ j ξ k (1 − | ξ | ) ‘ for nonnegative integers j, k, ‘ . Lemma 4.1.
We have h C sω α ( e ‘k + m,k ) i ( z ) = 0 if m > h C sω α ( e ‘k,k + m ) i ( z ) = γ ωk,s z m +1 if m ≥ . (4.1) where γ ωk,s stands for γ ωk,s = Z t k (1 − t ) s ω ( t ) dt. (4.2) Proof.
Notice first that for any z ∈ D c and ξ ∈ D, we have ξ/z ∈ D . Then, byexpanding the kernel function, we obtain h C sω α ( e sjk ) i ( z ) = 1 π + ∞ X l =0 z l +1 Z D ξ j + l ( ξ ) k (1 − | ξ | ) s ω (cid:16) | ξ | (cid:17) dλ ( ξ )= + ∞ X l =0 z l +1 (cid:18)Z t k (1 − t ) s ω ( t ) dt (cid:19) δ j + l,k = ε k − j γ ωk,s z k − j +1 , (4.3)where ε p = ( p ≥
00 if p < . (4.4)The last equalities follow by the use of polar coordinates ξ = re iθ , the fact that R π e i ( m − n ) θ dθ = 2 πδ n,m and the change of variable r = t , keeping in mind thedefinition of γ ωk,s and ε p given through (4.2) and (4.4), respectively. (cid:3) Accordingly, it is clear from Lemma 4.1 that the holomorphic functions C sω α ( e sjk )belong to Ker ( C sω α ), for any s and any j > k . Moreover, we assert the following Proposition 4.2.
The function C sω α ( e sjk ) belongs to L ,B ( D c ) and its square normis given by (cid:13)(cid:13)(cid:13) C sω α ( e sjk ) (cid:13)(cid:13)(cid:13) B = πε k − j ( γ ωk,s ) Z B (1 /u ) u k − j +1 du. Moreover, the system (cid:16) C sω α ( e sjk ) (cid:17) j,s is orthogonal in L ,B ( D c ) for every fixed k .Proof. For the proof, we need only to compute D C sω α ( e sjk ) , C sω α ( e rmn ) ω α E B for arbi-trary m, n, j, k . Indeed, by (4.3), we get D C s ( e sjk ) , C s ( e rmn ) E B = ε k − j ε n − m γ ωk,s γ ωn,r Z D c z k − j +1 z n − m +1 B ( | z | ) dλ ( z )= πε k − j γ ωk,s γ ωn,r δ k − j,n − m Z + ∞ t k − j +1 B ( t ) dt = πε k − j γ ωk,s γ ωn,r δ k − j,n − m Z u k − j − B (1 /u ) du. (4.5)This proves the second assertion in 1), while the first assertion follows as particularcase by taking m = j . The result in 2) is exactly (4.5) with m = j and n = k . (cid:3) Remark 4.3.
In view of (4.5) , it is clear that the family (cid:16) C sω α ( e sjk ) (cid:17) j,k,s is not. Using Lemma 4.1, we give the explicit action of C sω α on the disc polynomials. Proposition 4.4.
Let γ > − . Then, for every nonnegative integers m, n , thereexists some constant c γ,ωm,n , depending in γ , ω , m and n , such that C sω α ( R γm,n )( z ) = c γ,ωm,n z m z n +1 . For the weight function ω ( t ) = ω α ( t ) = (1 − t ) α , it is given explicitly by c γ,ω α m,n = ε n − m ( γ − α ) m n !( α + 1) n +1 ( γ + 1) m . R. EL HARTI, A. ELKACHKOURI, A. GHANMI
Proof.
By setting c γ,jm,n = ( − j m ! n !( γ + 1) j j !( m − j )!( n − j )! , we can rewrite the disc polynomials (2.1) by mean of the generic elements e jn − j,m − j as R γm,n ( z, ¯ z ) = m ∧ n X j =0 c γ,jm,n e jm − j,n − j ( z, ¯ z ) . Subsequently, the linearity of the weighted Cauchy transform and Lemma 4.1 showthat C sω α ( R γm,n )( ξ ) = ε n − m m X j =0 c γ,jm,n γ ωn − j,j z n − m +1 = c γ,ωm,n z m z n +1 , where the quantity γ ω α n − j,j , for ω ( t ) := ω α ( t ) = (1 − t ) α , reduces to a beta function, γ ω α n − j,j = Z t n − j (1 − t ) α + j dt = ( n − j )!( α + 1) j ( α + 1) n +1 . Therefore, the computation of the involved finite sum, that we denote S γ,αm,n , it turnsout to be the value of a Gauss hypergeometric function at 1. Indeed, we have S γ,αm,n = n !( α + 1) n +1 m X j =0 ( − m ) j ( α + 1) j ( γ + 1) j j != n !( α + 1) n +1 2 F − m, α + 1 γ + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ! . (4.6)By mean of Chu–Vandermonde identity F − m, bc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ! = ( c − b ) m ( c ) m , the explicit expression of the constant c γ,ωm,n , for ω ( t ) = ω α ( t ), reduces to c γ,ω α m,n = ε n − m ( γ − α ) m n !( α + 1) n +1 ( γ + 1) m . (cid:3) Subsequently, it is clear that the following assertions hold trues1) The range of C ω α s restricted to A ,γn ( D ) is finite dimensional vector space withdimension do not exceed n + 1, since R γm,n ( ξ, ξ ) form an orthogonal basis ofthe true weighted Bergman spaces [6].2) R γm,n ∈ ker( C sω α ), for any m > n.
3) We have C sω α ( R γm,n ) = const. C s ( e sn,m ) , and for fixed n and varying m =0 , , , · · · , they form an orthogonal system of is holomorphic functions in L ,B ( D c ).4) For α = γ , the involved constant is exactly zero for any m >
0, while for m = 0, it reduces to C sω α ( R γ ,n )( ξ ) = n !( γ + 1) n +1 z n +1 . With the material presented in this section we are now able to prove our mainresult.
Notice first that since we are placed in the case of ω = ω α , we have to assumethat α > ( γ − / > − C ω α s , thefiniteness of the weight function ω γ and therefore the fact that the disc polynomials isan orthogonal basis of L ,γ ( D ). According to the explicit expression of the constant c γ,ω α m,n in Proposition 4.4, it is clear that C ω α s R γm,n = 0 if and only if n < m and( γ − α ) m = 0 with m > α − γ . In particular, Span {R γm,n ; m < n } ⊂ ker C ω α s . Forthe determination of dim( C ω α s ( A ,γn ( D ))) ≤ n + 1, two cases are to be distinguished γ − α ∈ Z − = { , , , · · · } and γ − α / ∈ Z − .Thus, if γ − α / ∈ Z − , then ( γ − α ) m is not zero for every nonnegative integer m ≥
0, and C ω α s R γm,n = 0 if and only if m > n . Subsequently, the restriction of thesolid weighted Cauchy transform C s to A ,γn ( D ) is spanned by z m z n + 1 ; m = 0 , , , · · · , n. So that the dimension of is infected by the weight functions and is equal to n + 1.Now, when γ − α ∈ Z − , we have C sω α ( R γm,n )( ξ ) =
0; if m > n ( γ − α ) m n !( α + 1) n +1 ( γ + 1) m z m z n +1 ; if m ≤ min( n, α − γ + 1)0; if α − γ + 1 ≤ m ≤ n. In this case, C sω α = 0 if and only if m ≤ min( n, α − γ + 1). Hence, it follows C sω α ( A ,γn ( D )) = Span (cid:26) z m z n +1 ; 0 ≤ m ≤ min( n, α − γ + 1) (cid:27) and ker C ω α s | A ,γn ( D ) = Span n R γm,n ; m ≥ min( n, α − γ + 1) o . Clearly the dimension of C sω α ( A ,γn ( D )) is finite and given by N = min( n, α − γ + 1) + 1 . Moreover, C sω α ( A ,γn ( D )) n is an increasing sequence of spaces. Remark 5.1.
For γ = α , the ranges C sω α ( A ,γn ( D )) n are of all of dimension one. Remark 5.2.
The case of γ > α > ( γ − / > − is clearly contained in the caseof γ − α / ∈ Z − . Wile the case of − < γ < α depends on the quantization of γ − α . References [1] Aharmim B., El Hamyani A., El Wassouli F., Ghanmi A., Generalized Zernike polynomials:operational formulae and generating functions. Integral Transforms Spec. Funct. 26 (2015),no. 6, 395–410.[2] Balk M.B., Polyanalytic functions. Mathematical Research, 63. Akademie-Verlag, Berlin, 1991.[3] Bell S.R., The Cauchy transform, potential theory and conformal mapping. Second edition.Chapman & Hall/CRC, Boca Raton, FL, 2016.[4] A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators. Proc. Nat. Acad.Sci. U.S.A. 74 (1977), no. 4, 1324?1327.[5] Dunkl C.F., The Poisson kernel for Heisenberg polynomials on the disk. Math. Z.1984;187(4):527–547.[6] El Harti R., ElKachkouri A.„ A. Ghanmi A., A note on weighted poly-Bergman spaces.arXiv:2008.12764 math.CV
R. EL HARTI, A. ELKACHKOURI, A. GHANMI [7] El Hamyani A., Ghanmi A., Intissar A., Generalized Zernike polynomials: Integral represen-tation and Cauchy transform. arXiv:1605.00281 math.CA[8] Isaev K. P., Yulmukhametov R. S., The Laplace transform of functionals on Bergman spaces.(Russian. Russian summary) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 1, 5?42; transla-tion in Izv. Math. 68 (2004), no. 1, 3?41[9] Koornwinder T.H., Two-variable analogues of the classical orthogonal polynomials. Theoryand application of special functions, R.A. Askey (ed.), Academic Press, New York, 1975:435–495.[10] Koornwinder T.H., Positivity proofs for linearization and connection coefficients of orthogonalpolynomials satisfying an addition formula. J Lond. Math. Soc. 1978;18(2):101–114.[11] V. I. Lutsenko, R. S. Yulmukhametov, Generalization of the Wiener-Paley theorem to func-tionals in Smirnov spaces, (Russian), Trudy Mat. Inst. Steklov, Vol. 200, 1991, 245?254.[12] Merenkov S.A., On the Cauchy transform of the Bergman space. Mat. Fiz. Anal. Geom. 7(2000), no. 1, 119–127.[13] Merenkov S.A., On the Cauchy transform of weighted Bergman space. Arxiv 2013.[14] Napalkov V.V., Jr. Yulmukhametov R.S., On the Cauchy transform of functionals on theBergman space (Russian); translated from Mat. Sb. 185 (1994), no. 7, 77–86 Russian Acad.Sci. Sb. Math. 82 (1995), no. 2,[15] Napalkov, V. V., Jr.; Youlmukhametov, R. S. Criterion of surjectivity of the Cauchy transformoperator on a Bergman space. Entire functions in modern analysis (Tel-Aviv, 1997), 261?267,Israel Math. Conf. Proc., 15, Bar-Ilan Univ., Ramat Gan, 2001. 327–336[16] Ramazanov A. K., Representation of the Space of Polyanalytic Functions as a Direct Sumof Orthogonal Subspaces. Application to Rational Approximations; Mathematical Notes, Vol.66, No. 5, 1999.[17] Ramazanov A. K., On the Structure of Spaces of Polyanalytic Functions. Mathematical Notes,vol. 72, no. 5, 2002, pp. 692?704.[18] G. Rozenblum, N. Vasilevski,Toeplitz operators in polyanalytic Bergman type spaces, Func-tional Analysis and Geometry. Selim Grigorievich Krein Centennial. Contemporary Math-ematics,733, AMS, (2019).[19] Vasilevski N.L.,
Poly-Fock spaces . Oper. Theory, Adv. App. 117 (2000) 371–386.
E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected]@um5.ac.ma