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BY 4.0 license Open Access Published by De Gruyter Open Access June 6, 2022

Cauchy-Riemann ̄∂-equations with some applications

Jie Xiao and Cheng Yuan
From the journal Complex Manifolds

Abstract

This paper shows that given 0 < p < 3 and a complex Borel measure µ on the unit disk 𝔻 the inhomogeneous Cauchy-Riemann ̄∂-equation

z¯u(z)=dμ(z)(2πi)-1dz¯dz− a complex Gauss curvature of the weighted disk (𝔻, µ) ᗄ z ∈ 𝔻,

has a distributional solution (initially defined on ̄𝔻 = 𝔻 ∪ 𝕋) u ∈ ℒ2,p(𝕋) (formed of: (i) Morrey’s space M2,0<p<1(𝕋); (ii) John-Nirenberg’s space BMO(𝕋) = 𝒧2,1(𝕋); (iii) Hölder-Lipschitz’s space CC0<p-12<1 (𝕋)), if and only if

𝔻¯z𝔻(1-zw¯)-1dμ¯(w) belongs to the analytic Campanato space ϱ𝒜p(𝔻),

thereby not only extending Carleson’s corona & Wolff’s ideal theorems to the algebra M ϱ𝒜p(𝔻) of all analytic pointwise multiplications of ϱ𝒜p(𝔻), but quadratically generalizing Brownawell’s result on Hilbert’s Nullstellensatz for the analytic polynomial class 𝒫(ℂ).

MSC 2010: 32W05; 11C08; 30H05; 30H80; 46J20; 30H25

References

[1] D. P. Banjade, Wolff’s ideal theorem on Qp spaces. Rocky Mountain J. Math. 49 (2019)2121-2133.Search in Google Scholar

[2] B. Berndtsson, Weighted estimates for ̄∂ in domains in ℂ. Duke Math. J. 66(1992)239-255.Search in Google Scholar

[3] B. Berndtsson, An Introduction to Things ̄∂. Lecture notes from a summer school in Park City 2008. IAS/Park City Math Ser 17. Amer. Math. Soc. Providence RI 2010.Search in Google Scholar

[4] W.D. Brownawell, Bounds for the degrees in the Nullstellensatz. Ann. Math. 126(1987)577-591.Search in Google Scholar

[5] L. Carleson, Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76(1962)547-559.Search in Google Scholar

[6] C. Cascante, J. Fabrega and J. M. Ortega, The corona theorem in weighted Hardy and Morrey spaces. Ann. Scuola. Norm. Sup. Pisa. 13 (2014) 579-607.Search in Google Scholar

[7] P. L. Duren, B.W. Romberg and A.L. Shield, Linear functionals on Hp spaces with 0 < p < 1. J. Reine Angew. Math. 238(1969)32-60.Search in Google Scholar

[8] C. Fefferman and E.M. Stein, Hp spaces of several variables. Acta Math. 129(1972)137-193.Search in Google Scholar

[9] J. Garnett, Bounded Analytic Functions. GTM 236, Springer, 2007.Search in Google Scholar

[10] D. Hilbert, Über die vollen Invariantesysteme. Math. Ann. 42(1893)313-373.Search in Google Scholar

[11] L. Hörmander, Generators for some rings of analytic functions. Bull. Amer. Math. Soc. 73(1967)943-949.Search in Google Scholar

[12] P. Jones, L estimates for the ̄∂ problem in a half-plane. Acta Math 150(1983)137-138.10.1007/BF02392970Search in Google Scholar

[13] J. Kollár, Sharp effective Nullstellensatz. J. Amer. Math. Soc. (1988)963-975.10.1090/S0894-0347-1988-0944576-7Search in Google Scholar

[14] S.G. Krantz and S.Y. Li, Some remarks on the corona problem on strongly pseudoconvex domains. Illinois J. Math. 39(1995)323-349.Search in Google Scholar

[15] S.G. Krantz and S.Y. Li, Explicit solutions for the corona problem with Lipschitz data in the polydisc. Pacific J. Math. 174(1996)443-458.Search in Google Scholar

[16] H. Kwon, A. Netyanun and T. Trent, An analytic approach to the degree bound in the Nullstellensatz. Proc. Amer. Math. Soc. 144(2016)1145-1152.Search in Google Scholar

[17] P. Li, J. Liu and Z. Lou, Integral operators on analytic Morrey spaces. Sci. China Math. 57(2014)1961-1974.Search in Google Scholar

[18] J. Liu and Z. Lou, Properties of analytic Morrey spaces and applications. Math. Nachr. 288(2015)1673-1693.Search in Google Scholar

[19] P.F. Menal, G.N. Monreal and A. Nicolau, Interpolating sequences for analytic self-mappings of the disc. Amer. J. Math. 133(2011)437-465.10.1353/ajm.2011.0012Search in Google Scholar

[20] J.M. Ortega and J. Fàbrega, Pointwise multipliers and corona type decomposition in BMOA. Ann. Inst. Fourier, Grenoble 46(1996)111-137.Search in Google Scholar

[21] C. Tong and C. Yuan, An integral operator preserving s-Carleson measure on the unit ball. Ann. Acad. Sci. Fenn. Math. 40(2015)361-373.Search in Google Scholar

[22] S.Treil, Estimates in the corona theorem and ideals of H: A problem of T. Wolff. J. d’Analyse Math. 87(2002)481-495.10.1007/BF02868486Search in Google Scholar

[23] J. Wang and J. Xiao, Analytic Campanato spaces by functionals and operators. J. Geom. Anal. 26(2016)2996-3018.10.1007/s12220-015-9658-7Search in Google Scholar

[24] J. Xiao, The ̄∂-problem for multipliers of the Sobolev space. Manuscripta Math. 97(1998)217-232.Search in Google Scholar

[25] J. Xiao, Hankel measures on Hardy space. Bull. Austral. Math. Soc. 62(2000)135-140.Search in Google Scholar

[26] J. Xiao, Pseudo-Carleson measures for weighted Bergman spaces. Michigan Math. J. 47(2000)447-452.Search in Google Scholar

[27] J. Xiao, Holomorphic Q Classes. Lecture Notes in Math. 1767, 2000.Search in Google Scholar

[28] J. Xiao, Geometric Qp Functions. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.Search in Google Scholar

[29] R. Zhao, Generalization of Schur’s test and its application to a class of integral operators on the unit ball of ℂn. Integr. Equ. Oper. Theory 82(2015)519-532.Search in Google Scholar

[30] K. Zhu, Operator Theory in Function Spaces. 2nd edition, Amer. Math. Soc. 2007.Search in Google Scholar

Received: 2022-01-28
Accepted: 2022-05-14
Published Online: 2022-06-06

© 2022 Jie Xiao et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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