Maximal estimate and integral operators in Bergman spaces with doubling measure

Pang, Changbao; Perälä, Antti; Wang, Maofa; Guo, Xin

doi:10.1090/proc/14927

Maximal estimate and integral operators in Bergman spaces with doubling measure
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by Changbao Pang, Antti Perälä, Maofa Wang and Xin Guo

Proc. Amer. Math. Soc. 151 (2023), 2881-2894

DOI: https://doi.org/10.1090/proc/14927

Published electronically: March 30, 2023

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Previous version: Original version posted March 30, 2023
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Abstract:

The boundedness of the maximal operator on the upper half-plane $\Pi ^{+}$ is established. Here $\Pi ^+$ is equipped with a positive Borel measure $d\omega (y)dx$ satisfying the doubling property $\omega ((0,2t))\leq C\omega ((0,t))$. This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces $A_{\omega }^{p}(\Pi ^{+})$.

References

Alexandru Aleman and Joseph A. Cima, An integral operator on $H^p$ and Hardy’s inequality, J. Anal. Math. 85 (2001), 157–176. MR 1869606, DOI 10.1007/BF02788078
Alexandru Aleman and Olivia Constantin, Spectra of integration operators on weighted Bergman spaces, J. Anal. Math. 109 (2009), 199–231. MR 2585394, DOI 10.1007/s11854-009-0031-2
Alexandru Aleman and Aristomenis G. Siskakis, An integral operator on $H^p$, Complex Variables Theory Appl. 28 (1995), no. 2, 149–158. MR 1700079, DOI 10.1080/17476939508814844
Alexandru Aleman and Aristomenis G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. MR 1481594, DOI 10.1512/iumj.1997.46.1373
A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Milutin R. Dostanić, Integration operators on Bergman spaces with exponential weight, Rev. Mat. Iberoam. 23 (2007), no. 2, 421–436. MR 2371433, DOI 10.4171/RMI/501
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Petros Galanopoulos, Daniel Girela, and José Ángel Peláez, Multipliers and integration operators on Dirichlet spaces, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1855–1886. MR 2746668, DOI 10.1090/S0002-9947-2010-05137-2
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Daniel Girela and José Ángel Peláez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (2006), no. 1, 334–358. MR 2264253, DOI 10.1016/j.jfa.2006.04.025
G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403–439. MR 1545260, DOI 10.1007/BF01180596
Lars Hörmander, $L^{p}$ estimates for (pluri-) subharmonic functions, Math. Scand. 20 (1967), 65–78. MR 234002, DOI 10.7146/math.scand.a-10821
Birgit Jacob, Jonathan R. Partington, and Sandra Pott, On Laplace-Carleson embedding theorems, J. Funct. Anal. 264 (2013), no. 3, 783–814. MR 3003737, DOI 10.1016/j.jfa.2012.11.016
Changbao Pang, Antti Perälä, and Maofa Wang, Embedding theorems and area operators on Bergman spaces with doubling measure, Complex Anal. Oper. Theory 15 (2021), no. 3, Paper No. 42, 24. MR 4227821, DOI 10.1007/s11785-021-01089-4
Jordi Pau, Integration operators between Hardy spaces on the unit ball of $\Bbb {C}^n$, J. Funct. Anal. 270 (2016), no. 1, 134–176. MR 3419758, DOI 10.1016/j.jfa.2015.10.009
Jordi Pau and José Ángel Peláez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights, J. Funct. Anal. 259 (2010), no. 10, 2727–2756. MR 2679024, DOI 10.1016/j.jfa.2010.06.019
Miroslav Pavlović and José Ángel Peláez, An equivalence for weighted integrals of an analytic function and its derivative, Math. Nachr. 281 (2008), no. 11, 1612–1623. MR 2462603, DOI 10.1002/mana.200510701
José Ángel Peláez and Jouni Rättyä, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. 227 (2014), no. 1066, vi+124. MR 3155774
José Ángel Peláez and Jouni Rättyä, Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), no. 1-2, 205–239. MR 3343875, DOI 10.1007/s00208-014-1108-5
Jouni Rättyä, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Austral. Math. Soc. 75 (2007), no. 3, 431–446. MR 2331020, DOI 10.1017/S0004972700039356
Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI 10.4064/sm-75-1-1-11
Kehe Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. MR 2311536, DOI 10.1090/surv/138