Abstract
We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.
Similar content being viewed by others
References
Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press (1989)
Abate, M., Saracco, A.: Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains. J. Lond. Math. Soc. 83(3), 587–605 (2011)
Balogh, Z.M., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75(3), 504–533 (2000)
Brudnyi, A.: Local inequalities for plurisubharmonic functions. Ann. Math. 149(2), 511–533 (1999)
Calzi, M., Peloso, M.M.: Carleson and reverse Carleson measures on homogeneous Siegel domains. Complex Anal. Oper. Theory 16(1), 4 (2022)
Carleman, T.: Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys., 26, (1939)
Čučković, Ž, Şahutoğlu, S., Zeytuncu, Y.E.: A local weighted Axler-Zheng theorem in \(\mathbb{C} ^n\). Pac. J. Math. 294(1), 89–106 (2018)
David, G., Li, L., Mayboroda, S.: Carleson measure estimates for the Green function. Arch. Ration. Mech. Anal. 243(3), 1525–1563 (2022)
Fricain, E., Hartmann, A., Ross, W.T.: A survey on reverse Carleson measures. In: Harmonic analysis operator theory, function theory, and applications, pp. 91–123. Theta, Bucharest (2017)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer (2001)
Hartmann, A., Kamissoko, D., Konate, S., Orsoni, M.-A.: Dominating sets in Bergman spaces and sampling constants. J. Math. Anal. Appl. 495(2), 124755 (2021)
Hörmander, L.: An introduction to complex analysis in several variables. Elsevier (1973)
Hörmander, L.: The analysis of linear partial differential operators I-IV. Grundlehren, vol. 257. Springer, USA (1983)
Hu, Z., Lv, X., Zhu, K.: Carleson measures and balayage for Bergman spaces of strongly pseudoconvex domains. Math. Nachr. 289(10), 1237–1254 (2016)
Kerzman, N.: The Bergman kernel function. Differentiability at the boundary. Math. Ann. 195, 149–158 (1972)
Kovrijkine, O.: Some results related to the Logvinenko–Sereda theorem. Proc. Am. Math. Soc. 129(10), 3037–3047 (2001)
Krantz, S.G.: Function theory of several complex variables, vol. 340. American Mathematical Soc., (2001)
Krantz, S.G., Ma, D.: Bloch functions on strongly pseudoconvex domains. Indiana Univ. Math. J. 37(1), 145–163 (1988)
Lebeau, G., Moyano, I.: Spectral inequalities for the Schrödinger operator. arXiv preprint arXiv:1901.03513, (2019)
Li, H.: BMO, VMO and Hankel operators on the Bergman space of strongly pseudoconvex domains. J. Funct. Anal. 106(2), 375–408 (1992)
Logunov, A., Malinnikova, E.: Quantitative propagation of smallness for solutions of elliptic equations. In Proceedings of the International Congress of Mathematicians-Rio de, vol. 2, pp 2357–2378. World Scientific, (2018)
Luecking, D.H.: Inequalities on Bergman spaces. Illinois J. Math. 25(1), 1–11 (1981)
Luecking, D.H.: Closed ranged restriction operators on weighted Bergman spaces. Pac. J. Math. 110(1), 145–160 (1984)
Luecking, D.H.: Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. Am. J. Math. 107(1), 85–111 (1985)
Nazarov, F., Sodin, M., Volberg, A.: Lower bounds for quasianalytic functions, I. How to control smooth functions. Math. Scand. 95(1), 59–79 (2004)
Peloso, M.M.: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains. Illinois J. Math. 38(2), 223–249 (1994)
Ronkin, L.I.: Introduction to the theory of entire functions of several variables. Translations of Mathematical Monographs, vol. 44. American Mathematical Soc., Providence, RI (1974)
Wang, Y., Xia, J.: Essential commutants on strongly pseudo-convex domains. J. Funct. Anal. 280(1), 108775 (2021)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Kristian Seip.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by NSF GRF, Grant Number DGE-1745038.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Green, A.W., Wagner, N.A. Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains. Constr Approx 59, 229–269 (2024). https://doi.org/10.1007/s00365-023-09639-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-023-09639-z