×

Generalized Hölder type spaces of harmonic functions in the unit ball and half space. (English) Zbl 1524.42045

In this paper, the authors study spaces of Hölder type functions harmonic in the unit ball and half-space with some smoothness conditions up to the boundary.
They provide characterizations of Hölder type space of harmonic functions in the unit ball with modulus of continuity and the variable exponent harmonic Hölder space in the unit ball with the continuity modulus. These characterizations are given in terms of the growth of the gradient of a function near the boundary of the unit ball. Furthermore, they extend the results to the case of Hölder type spaces of harmonic functions in the half-space.

MSC:

42B35 Function spaces arising in harmonic analysis
46E15 Banach spaces of continuous, differentiable or analytic functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arsenović, M.; Kojić, V.; Mateljević, M., On Lipschitz continuity of harmonic quasiregular maps on the unit ball in \(\mathbb R^n\), Ann. Acad. Sci. Fenn., Math. 33 (2008), 315-318 · Zbl 1140.31003
[2] Axler, S.; Bourdon, P.; Ramey, W., Harmonic Function Theory, Graduate Texts in Mathematics 137, Springer, New York (2001) · Zbl 0959.31001 · doi:10.1007/b97238
[3] Blumenson, L. E., A derivation of \(n\)-dimensional spherical coordinates, Am. Math. Mon. 67 (1960), 63-66 · doi:10.2307/2308932
[4] Chacón, G. R.; Rafeiro, H., Variable exponent Bergman spaces, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 105 (2014), 41-49 · Zbl 1288.30059 · doi:10.1016/j.na.2014.04.001
[5] Chacón, G. R.; Rafeiro, H., Toeplitz operators on variable exponent Bergman spaces, Mediterr. J. Math. 13 (2016), 3525-3536 · Zbl 1354.30049 · doi:10.1007/s00009-016-0701-0
[6] Cruz-Uribe, D.; Fiorenza, A., Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg (2013) · Zbl 1268.46002 · doi:10.1007/978-3-0348-0548-3
[7] Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Berlin (2011) · Zbl 1222.46002 · doi:10.1007/978-3-642-18363-8
[8] Duren, P.; Schuster, A., Bergman Spaces, Mathematical Surveys and Monographs 100, American Mathematical Society, Providence (2004) · Zbl 1059.30001 · doi:10.1090/surv/100
[9] Hedenmalm, H.; Korenblum, B.; Zhu, K., Theory of Bergman Spaces, Graduate Texts in Mathematics 199, Springer, New York (2000) · Zbl 0955.32003 · doi:10.1007/978-1-4612-0497-8
[10] Karapetyants, A.; Rafeiro, H.; Samko, S., Boundedness of the Bergman projection and some properties of Bergman type spaces, Complex Anal. Oper. Theory 13 (2019), 275-289 · Zbl 1421.30070 · doi:10.1007/s11785-018-0780-y
[11] Karapetyants, A.; Samko, S., Spaces \(BMO_{p(\cdot)}(\Bbb D)\) of a variable exponent \(p(z)\), Georgian Math. J. 17 (2010), 529-542 · Zbl 1201.30072 · doi:10.1515/gmj.2010.028
[12] Karapetyants, A.; Samko, S., Mixed norm Bergman-Morrey-type spaces on the unit disc, Math. Notes 100 (2016), 38-48 · Zbl 1364.30063 · doi:10.1134/S000143461607004X
[13] Karapetyants, A.; Samko, S., Mixed norm variable exponent Bergman space on the unit disc, Complex Var. Elliptic Equ. 61 (2016), 1090-1106 · Zbl 1351.30042 · doi:10.1080/17476933.2016.1140750
[14] Karapetyants, A.; Samko, S., Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces, Fract. Calc. Appl. Anal. 20 (2017), 1106-1130 · Zbl 1386.30052 · doi:10.1515/fca-2017-0059
[15] Karapetyants, A.; Samko, S., On boundedness of Bergman projection operators in Banach spaces of holomorphic functions in half-plane and harmonic functions in half-space, J. Math. Sci., New York 226 (2017), 344-354 · Zbl 1386.30051 · doi:10.1007/s10958-017-3538-6
[16] Karapetyants, A.; Samko, S., Generalized Hölder spaces of holomorphic functions in domains in the complex plane, Mediterr. J. Math. 15 (2018), Paper No. 226, 17 pages · Zbl 1411.30039 · doi:10.1007/s00009-018-1272-z
[17] Karapetyants, A.; Samko, S., On mixed norm Bergman-Orlicz-Morrey spaces, Georgian Math. J. 25 (2018), 271-282 · Zbl 1392.30023 · doi:10.1515/gmj-2018-0027
[18] Kokilashvili, V.; Meskhi, A.; Rafeiro, H.; Samko, S., Integral Operators in Non-Standard Function Spaces. Volume 1. Variable Exponent Lebesgue and Amalgam Spaces, Operator Theory: Advances and Applications 248, Birkhäuser/Springer, Basel (2016) · Zbl 1385.47001 · doi:10.1007/978-3-319-21015-5
[19] Kokilashvili, V.; Meskhi, A.; Rafeiro, H.; Samko, S., Integral Operators in Non-Standard Function Spaces. Volume 2. Variable Exponent Hölder, Morrey-Campanato and Grand Spaces, Operator Theory: Advances and Applications 249, Birkhäuser/Springer, Basel (2016) · Zbl 1367.47004 · doi:10.1007/978-3-319-21018-6
[20] Zhu, K., Spaces of Holomorphic Functions in the Unit Ball, Graduate texts in Mathematics 226, Springer, New York (2005) · Zbl 1067.32005 · doi:10.1007/0-387-27539-8
[21] Zhu, K., Operator Theory in Function Spaces, Mathematical Surveys and Monographs 138, American Mathematical Society, Providence (2007) · Zbl 1123.47001 · doi:10.1090/surv/138
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.