We characterize the boundedness of the radial averaging operatorTω, induced by a radial weightωon the unit discD, from ApνtoLpν, where 0 p 8andνP pD. We also establish two characterizations of the weak-type inequality
ηptzPD:|Tωpfqpzq| ¥λuq Àλp}f}pLp
ν, λ¡0, for arbitrary radial weightsω,νandη.
It is worth noting that, under the hypothesesω,ν PRandηP qD, the averaging operatorTω and the maximal Bergman projection
Pωpfqpzq
»
D
fpζq|Bzωpζq|ωpζqdApζq, zPD,
are simultaneously bounded from Lpνto Lηpby Theorems 4.2 and 5.18. Moreover, in the particular caseω 1 withν P R,Tω : Aνp Ñ Lpν is bounded if and only if the Bergman projectionPω is bounded onLνp[58]. Therefore the averaging operatorTω, which is interesting in its own right, is closely related to the weighted Bergman projectionPω and might be further used as a model for its study. With this aim and that of studying the difference with respect to their action on Lebesgue spaces, we consider the averaging operatorTω acting fromAνptoLpνwhenνP pD.
Recall the characterization ofq-Carleson measures for weighted Bergman spaces given in Theorem 2.11. By denoting
dµp,ω,νpzq pωpzqp1ωpzq we state the following result.
Theorem 5.21. [33, Theorem 1.1] Let0 p 8,νP pDandωbe a radial weight. Then the following statements are equivalent:
(i) TωN :AνpÑLpνis bounded;
The proof of the necessity of the conditionDpppω,νq 8differs from the analo-gous part of Theorem 4.2 in that the test functions cannot be chosen to be indicators, and we end up using derivatives of Bergman kernels Baνinstead. This requires the use of Theorem 2.4 along with other estimates given in [52]. Other tools needed in this direction of the proof are Hardy-Littlewood inequalities and smooth polynomi-als related to Hadamard products. Our approach actually yields a sharp necessary condition in a more general case.
Proposition 5.22. [33, Proposition 2.2] Let 0 p ¤q 8, ν P pDand ω,η be radial
Another difference between analytic and classical cases arises in determining a sufficient condition for the boundedness. In the analytic case, Carleson measures prove to be an adequate tool. We also observe that the condition Dpppω,νq 8 is self-improving in the sense that if Dpppω,νq 8, thenDppεpω,νq 8 for some εεpp,ω,νq ¡0, see [33, Lemma 2.5] for details.
Using Theorems 5.21 and 4.2 along with [52, Theorem 11], we establish the fol-lowing result.
The statement in Corollary 5.23 fails without any local regularity hypotheses on the weights. This is no great surprise since, by Theorem 4.2,ωneeds to be absolutely continuous with respect toνforTω to be bounded onLpν, see [33, Corollary 3.1].
Finally, we present the following result regarding the weak-type inequality for the operatorTω.
(iii) Mp,εpω,ν,ηq sup
0¤r 1
ωprqp ε
»r
0
sηpsq ωpsqp p ε ds
1p»1
r
ωpsq sνpsq
p1
sνpsqds 1
p1
8 for some (equivalently for each)ε¡0.
Moreover, for each fixedε¡0, }Tω}Lp
νÑLηp,8 Nppω,ν,ηq Mp,εpω,ν,ηq.
The proof of Theorem 5.24 follows the leading idea in Theorem 4.3. However, due to the presence of the weight in the operatorTω and our function spaces being of the disc variant, a number of non-trivial modifications and additions are needed.
The weak and strong inequalities forTω are, in general, not equivalent: for each 1 p 8, there are weightsω,ν andη such thatTω : Lνp ÑLηp,8 is bounded but Tω :LνpÑLηpis not, see the discussion after [33, Corollary 10]. However, combining our results with suitable restrictions on the weights yields the following.
Corollary 5.25. [33, Corollary 1.4] Let1 p 8,ωP pDandνPD. Then Tω :LνpÑLνp
is bounded if and only if Tω :ApνÑLνpis bounded and Tω :LνpÑLνp,8is bounded.
BIBLIOGRAPHY
[1] A. Aleman, “A class of integral operators on spaces of analytic functions,” in Topics in complex analysis and operator theory, Proceedings of the Winter School on Complex Analysis and Operator Theory held in Málaga, February 5-9, 2006 (2007), pp. 3–30.
[2] A. Aleman and J. A. Cima, “An integral operator onHpand Hardy’s inequal-ity,”J. Anal. Math.85, 157–176 (2001).
[3] A. Aleman, S. Pott, and M. Reguera, “Sarason conjecture on the Bergman space,”Int. Math. Res. Not. IMRN4320–4349 (2017).
[4] A. Aleman and C. Sundberg, “Zeros of functions in weighted Bergman spaces,”
inLinear and complex analysis, Vol. 226, Amer. Math. Soc. Transl. Ser. 2 (2009), pp. 1–10.
[5] K. Andersen and B. Muckenhoupt, “Weighted weak type inequalities with applications to Hilbert transforms and maximal functions,” Studia Math. 72, 9–26 (1982).
[6] D. Bekollé, “Inégalités á poids pour le projecteur de Bergman dans la boule unité deCn [Weighted inequalities for the Bergman projection in the unit ball ofCn],”Studia Math.71, 305–323 (1981/82).
[7] D. Bekollé and A. Bonami, “Inégalités á poids pour le noyau de Bergman,”
C. R. Acad. Sci. Paris Sér. A-B286, 775–778 (1978).
[8] L. Bernal, “On growthk-order of solutions of a complex homogeneous linear differential equation,”Proc. Amer. Math. Soc.101, 317–322 (1987).
[9] A. L. Bernardis, F. J. Martín-Reyes, and P. O. Salvador, “Weighted inequalities for Hardy-Steklov Operators,”Canad. J. Math.59, 276–295 (2007).
[10] L. Carleson, “Interpolations by bounded analytic functions and the corona problem,”Ann. of Math. (2)76, 547–559 (1962).
[11] I. Chyzhykov, G. Gundersen, and J. Heittokangas, “Linear differential equa-tions and logarithmic derivative estimates,”Proc. London Math. Soc. (3)86, 735–
754 (2003).
[12] O. Constantin, “Carleson embeddings and some classes of operators on weighted Bergman spaces,”J. Math. Anal. Appl.365, 668–682 (2010).
[13] J. Duoandikoetxea, F. J. Martín-Reyes, and S. Oombrosi, “Calderón weights as Muckenhoupt weights,”Indiana Univ. Math. J.62, 891–910 (2013).
[14] P. Duren,Theory of HpSpaces, Vol. 38 ofPure and Applied Mathematics(Academic Press, New York, 1970).
[15] M. Essén and H. Wulan, “On analytic and meromorphic functions and spaces ofQK-type,”Illinois J. Math.46, 1233–1258 (2002).
[16] A. Gogatishvili and J. Lang, “The generalized Hilbert operator with kernel and variable integral limits in Banach spaces,”J. Inequal. Appl.4, 1–16 (1999).
[17] J. Gröhn, “New applications of Picard’s successive approximations,” Bull. Sci.
Math.135, 475–487 (2011).
[18] H. Hedenmalm, “A factoring theorem for a weighted Bergman space,”Algebra i Analiz4, 167–176 (1992), translation in St. Petersburg Math. J. 4 (1993), no. 1, 163–174.
[19] J. Heittokangas, On complex differential equations in the unit disc, PhD thesis (University of Joensuu, 2000).
[20] J. Heittokangas, R. Korhonen, and J. Rättyä, “Growth estimates for solutions of linear complex differential equations,”Ann. Acad. Sci. Fenn. Math.29, 233–246 (2004).
[21] J. Heittokangas, R. Korhonen, and J. Rättyä, “Fast growing solutions of linear differential equations in the unit disc,”Results Math.49, 265–278 (2006).
[22] J. Heittokangas, R. Korhonen, and J. Rättyä, “Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space,” Nagoya Math. J.
187, 91–113 (2007).
[23] J. Heittokangas, R. Korhonen, and J. Rättyä, “Linear differential equations with coefficients in weighted Bergman and Hardy spaces,” Trans. Amer. Math. Soc.
360, 1035–1055 (2008).
[24] J. Heittokangas, R. Korhonen, and J. Rättyä, “Growth estimates for solutions of nonhomogeneous linear complex differential equations,”
Ann. Acad. Sci. Fenn. Math.34, 145–156 (2009).
[25] C. Horowitz, “Zeros of functions in the Bergman spaces,” Duke Math. J. 41, 693–710 (1974).
[26] C. Horowitz, “Factorization theorems for functions in the Bergman spaces,”
Duke Math. J.44, 201–213 (1977).
[27] C. Horowitz, “Some conditions on Bergman space zero sets,”J. Anal. Math.62, 323–348 (1994).
[28] C. Horowitz, “Zero sets and radial zero sets in function spaces,”J. Anal. Math.
65, 145–159 (1995).
[29] C. Horowitz and Y. Schnaps, “Factorization of functions in weighted Bergman spaces,”Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)29, 891–903 (2000).
[30] J.-M. Huusko, T. Korhonen, and A. Reijonen, “Linear differential equations with solutions in the growth space H8ω,” Ann. Acad. Sci. Fenn. Math. 41, 399–
416 (2016).
[31] T. Hytönen, “The sharp weighted bound for general Calderón-Zygmund oper-ators,”Ann. of Math. (2)175, 1473–1506 (2012).
[32] B. Korenblum, “An extension of the Nevanlinna theory,” Acta Math.135, 187–
219 (1975).
[33] T. Korhonen, J. A. Peláez, and J. Rättyä, “Radial averaging op-erator acting on Bergman and Lebesgue spaces,” Forum Math. (2019), https://doi.org/10.1515/forum-2018-0293.
[34] T. Korhonen, J. A. Peláez, and J. Rättyä, “Radial two weight inequality for maximal Bergman projection induced by a regular weight,”submitted, preprint available at https://arxiv.org/abs/1805.01256(2019).
[35] T. Korhonen and J. Rättyä, “Zero sequences, factorization and sampling mea-sures for weighted Bergman spaces,”Math. Z.291, 1145–1173 (2019).
[36] M. Lacey, E. Sawyer, S. Chun-Yen, and I. Uriarte-Tuero, “Two-weight inequality for the Hilbert transform: a real variable characterization, I.,”Duke Math. J.163, 2795–2820 (2014).
[37] M. Lacey, E. Sawyer, S. Chun-Yen, I. Uriarte-Tuero, and B. D. Wick, “Two Weight Inequalities for the Cauchy Transform fromRtoC ,” (2019), preprint available at https://arxiv.org/abs/1310.4820.
[38] I. Laine, Nevanlinna Theory and Complex Differential Equations, Vol. 15 of De Gruyter Studies in Mathematics(Walter de Gruyter & Co., Berlin, 1993).
[39] H. Li and H. Wulan, “Linear differential equations with solutions in the QK
spaces,”J. Math. Anal. Appl.375, 478–489 (2011).
[40] D. Luecking, “Inequalities on Bergman spaces,”Illinois J. Math.25, 1–11 (1981).
[41] D. Luecking, “Closed ranged restriction operators on weighted Bergman spaces,”Pacific J. Math.110, 145–160 (1984).
[42] D. Luecking, “Forward and Reverse Carleson Inequalities for Functions in Bergman Spaces and Their Derivatives,”Amer. J. Math.107, 85–111 (1985).
[43] D. Luecking, “Representation and Duality in Weighted Spaces of Analytic Functions,”Indiana Univ. Math. J.34, 319–336 (1985).
[44] D. Luecking, “Zero Sequences for Bergman Spaces,” Complex Variables Theory Appl.30, 345–362 (1996).
[45] D. Luecking, “Sampling measures for Bergman spaces on the unit disk,”
Math. Ann.316, 659–679 (2000).
[46] B. Muckenhoupt, “Hardy’s inequality with weights,” Studia Math. 44, 31–38 (1972), Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity.
[47] J. Pau and R. Zhao, “Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball,”Math. Ann.363, 363–383 (2015).
[48] J. Pau, R. Zhao, and K. Zhu, “Weighted BMO and Hankel operators between Bergman spaces,”Indiana Univ. Math. J.65, 1639–1673 (2016).
[49] J. A. Peláez, “Small weighted Bergman spaces,” in Proceedings of the summer school in complex and harmonic analysis, and related topics(2016), pp. 29–98.
[50] J. A. Peláez and J. Rättyä, “Weighted Bergman spaces induced by rapidly in-creasing weights,”Mem. Amer. Math. Soc.227, vi+124 pp (2014).
[51] J. A. Peláez and J. Rättyä, “Embedding Theorems for Bergman Spaces via Harmonic Analysis,”Math. Ann.362, 205–239 (2015).
[52] J. A. Peláez and J. Rättyä, “Two weight inequality for Bergman projection,” J.
Math. Pures Appl. (9)105, 102–130 (2016).
[53] J. A. Peláez and J. Rättyä, “Bergman projection induced by radial weight,”
(2019), preprint available at https://arxiv.org/abs/1902.09837.
[54] J. A. Peláez, J. Rättyä, and K. Sierra, “Embedding Bergman spaces into tent spaces,”Math. Z.281, 1215–1237 (2015).
[55] J. A. Peláez, J. Rättyä, and K. Sierra, “Berezin transform and Toeplitz operators on weighted Bergman spaces induced by regular weights,”J. Geom. Anal. 28, 656–687 (2018).
[56] J. A. Peláez, J. Rättyä, and B. D. Wick, “Bergman projection induced by kernel with integral representation,” (2019), to appear in J. Anal. Math., preprint available at https://arxiv.org/abs/1606.00718.
[57] V. V. Peller, Hankel operators and their applications (Springer-Verlag, New York, 2003).
[58] F. Pérez-González and J. Rättyä, “Derivatives of inner functions in weighted Bergman spaces and the Schwarz-Pick lemma,” Proc. Amer. Math. Soc. 145, 2155–2166 (2017).
[59] C. Pommerenke, “On the mean growth of the solutions of complex linear differential equations in the disk,”Complex Var. Theory Appl.1, 23–38 (1982/83).
[60] E. Sawyer, “Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator,”Trans. Amer. Math. Soc.281, 329–337 (1984).
[61] K. Seip, “On a theorem of Korenblum,”Ark. Mat.32, 237–243 (1994).
[62] K. Seip, “On Korenblum’s density condition for the zero sequences ofAα,”J.
Anal. Math.67, 307–322 (1995).
[63] A. Shields and D. Williams, “Bounded projections and the growth of harmonic conjugates in the unit disc,”Michigan Math. J.29, 3–25 (1982).
[64] K. Zhu, Operator theory in function spaces(Marcel Dekker, Inc., New York, 1990).
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DISSERTATIONS | TANELI KORHONEN | ON LINEAR OPERATORS AND THEIR APPLICATIONS IN... | No 344
TANELI KORHONEN
ON LINEAR OPERATORS AND THEIR APPLICATIONS IN COMPLEX FUNCTION SPACES OF THE UNIT DISC
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This thesis contains several new results on linear operators acting on spaces of
complex-valued functions and their applications to the study of analytic functions. Two weight inequalities for maximal Bergman projection and a radial averaging operator are discussed.
We apply Carleson measures and boundedness of certain operators in the study of weighted Bergman spaces. Operator theoretic approach
yields results on the growth of solutions of linear differential equations in the unit disc.