×

Harmonic Besov spaces on the ball. (English) Zbl 1354.31005

The authors study the properties of Besov spaces of harmonic functions on the unit ball \(\mathbb{B}\). They introduce the (two-parameter) harmonic Besov spaces \(b_q^p\) and they show several equivalent characterizations of such spaces. They prove that the spaces \(b^2_q\) are reproducing kernel Hilbert spaces. Then they consider the Bergman-Besov projections from a certain (weighted) Lebesgue spaces \(L^p_q\) onto \(b^p_q\) and show necessary and sufficient conditions for their boundedness.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions

Software:

DLMF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 1. S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd edn., Graduate Texts in Mathematics, Vol. 137 (Springer, New York, 2001). genRefLink(16, ’S0129167X16500701BIB001’, ’10.1007 · Zbl 0959.31001
[2] 2. F. Beatrous and J. Burbea, Holomorphic Sobolev spaces on the ball, Dissertationes Math.276 (1989) 57 pp. · Zbl 0691.46024
[3] 3. J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren der Mathematischen Wissenschaften, Vol. 223 (Springer, Berlin, 1976). genRefLink(16, ’S0129167X16500701BIB003’, ’10.1007
[4] 4. O. Blasco and S. Pérez-Esteva, Lp continuity of projectors of weighted harmonic Bergman spaces, Collect. Math.51 (2000) 49-58. · Zbl 0947.31001
[5] 5. B. R. Choe, H. Koo and K. Na, Positive Toeplitz operators of Schatten-Herz type, Nagoya Math. J.185 (2007) 31-62. genRefLink(16, ’S0129167X16500701BIB005’, ’10.1017
[6] 6. B. R. Choe, H. Koo and K. Nam, Optimal norm estimate of operators related to the harmonic Bergman projection on the ball, Tohoku Math. J.62 (2010) 357-374. genRefLink(16, ’S0129167X16500701BIB006’, ’10.2748 · Zbl 1203.31006
[7] 7. B. R. Choe, H. Koo and H. Yi, Derivatives of harmonic Bergman and Bloch functions on the ball, J. Math. Anal. Appl.260 (2001) 100-123. genRefLink(16, ’S0129167X16500701BIB007’, ’10.1006
[8] 8. B. R. Choe, H. Koo and H. Yi, Projections for harmonic Bergman spaces and applications, J. Funct. Anal.216 (2004) 388-421. genRefLink(16, ’S0129167X16500701BIB008’, ’10.1016 · Zbl 1066.31003
[9] 9. B. R. Choe and H. Y. J. Lee, Note on atomic decompositions of harmonic Bergman functions, in Complex Analysis and Its Applications, OCAMI Studies, Vol. 2 (Osaka Municipal University, Osaka, 2007), pp. 11-24. · Zbl 1154.47019
[10] 10. B. R. Choe, Y. J. Lee and K. Na, Toeplitz operators on harmonic Bergman spaces, Nagoya Math. J.174 (2004) 165-186. genRefLink(16, ’S0129167X16500701BIB010’, ’10.1017
[11] 11. E. S. Choi and K. Na, Characterizations of the harmonic Bergman space on the ball, J. Math. Anal. Appl.353 (2009) 375-385. genRefLink(16, ’S0129167X16500701BIB011’, ’10.1016 · Zbl 1162.31003
[12] 12. R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in Lp, Astérisque77 (1980) 12-66. · Zbl 0472.46040
[13] 13. S. Gergün, H. T. Kaptanoğlu and A. E. Üreyen, Reproducing kernels for harmonic Besov spaces on the ball, C. R. Math. Acad. Sci. Paris347 (2009) 735-738. genRefLink(16, ’S0129167X16500701BIB013’, ’10.1016 · Zbl 1179.31003
[14] 14. M. Jevtić and M. Pavlović, Harmonic Bergman functions on the unit ball in \(\mathbb{R}\)n, Acta Math. Hungar.85 (1999) 81-96. genRefLink(16, ’S0129167X16500701BIB014’, ’10.1023
[15] 15. M. Jevtić and M. Pavlović, Harmonic Besov spaces on the unit ball in \(\mathbb{R}\)n, Rocky Mountain J. Math.31 (2001) 1305-1316. genRefLink(16, ’S0129167X16500701BIB015’, ’10.1216
[16] 16. H. Kang and H. Koo, Estimates of the harmonic Bergman kernel on smooth domains, J. Funct. Anal.185 (2001) 220-239. genRefLink(16, ’S0129167X16500701BIB016’, ’10.1006
[17] 17. H. T. Kaptanoğlu, Bergman projections on Besov spaces on balls, Illinois J. Math.49 (2005) 385-403. genRefLink(128, ’S0129167X16500701BIB017’, ’000233210500004’); · Zbl 1079.32004
[18] 18. H. T. Kaptanoğlu, Reproducing kernels and radial differential operators for holomorphic and harmonic Besov spaces on unit balls: A unified view, Comput. Methods Funct. Theory10 (2010) 483-500. genRefLink(16, ’S0129167X16500701BIB018’, ’10.1007
[19] 19. H. T. Kaptanoğlu and A. E. Üreyen, Analytic properties of Besov spaces via Bergman projections, Contemp. Math.455 (2008) 169-182. genRefLink(16, ’S0129167X16500701BIB019’, ’10.1090
[20] 20. D. Karp, private communication (2010).
[21] 21. H. Koo, K. Nam and H. Yi, Weighted harmonic Bergman functions on half-spaces, J. Korean Math. Soc.42 (2005) 975-1002. genRefLink(16, ’S0129167X16500701BIB021’, ’10.4134 · Zbl 1136.31003
[22] 22. E. Ligocka, The Sobolev spaces of harmonic functions, Studia Math.84 (1986) 79-87. genRefLink(128, ’S0129167X16500701BIB022’, ’A1986G312100005’); · Zbl 0627.46033
[23] 23. E. Ligocka, Estimates in Sobolev norms s for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions, Studia Math.86 (1987) 255-271. genRefLink(128, ’S0129167X16500701BIB023’, ’A1987M068800005’); · Zbl 0642.46035
[24] 24. E. Ligocka, On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in \(\mathbb{R}\)n, Studia Math.87 (1987) 23-32. genRefLink(128, ’S0129167X16500701BIB024’, ’A1987M571200003’); · Zbl 0658.31006
[25] 25. E. Ligocka, On the space of Bloch harmonic functions and interpolation of spaces of harmonic and holomorphic functions, Studia Math.87 (1987) 223-238. genRefLink(128, ’S0129167X16500701BIB025’, ’A1987M781700003’); · Zbl 0657.31006
[26] 26. E. Ligocka, Corrigendum to the paper ”On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in \(\mathbb{R}\)n”, Studia Math.101 (1992) 319. · Zbl 0810.31003
[27] 27. C. W. Liu and J. H. Shi, Invariant mean-value property and -harmonicity in the unit ball of \(\mathbb{R}\)n, Acta Math. Sin.19 (2003) 187-200. genRefLink(16, ’S0129167X16500701BIB027’, ’10.1007
[28] 28. C. Liu, J. Shi and G. Ren, Duality for harmonic mixed-norm spaces in the unit ball of \(\mathbb{R}\)n, Ann. Sci. Math. Québec25 (2001) 179-197. · Zbl 1005.31003
[29] 29. J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math.125 (1998) 25-35. genRefLink(16, ’S0129167X16500701BIB029’, ’10.1007
[30] 30. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University, New York, 2010). · Zbl 1198.00002
[31] 31. R. Otáhalová, Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball, Proc. Amer. Math. Soc.136 (2008) 2483-2492. genRefLink(16, ’S0129167X16500701BIB031’, ’10.1090
[32] 32. S. Pérez-Esteva, Duality on vector-valued weighted harmonic Bergman spaces, Studia Math.118 (1996) 37-47. genRefLink(128, ’S0129167X16500701BIB032’, ’A1996UB70300005’); · Zbl 0854.46022
[33] 33. G. Ren, Harmonic Bergman spaces with small exponents in the unit ball, Collect. Math.53 (2003) 83-98. · Zbl 1029.46019
[34] 34. G. Ren and U. Kähler, Weighted harmonic Bloch spaces and Gleason’s problem, Complex Var. Theory Appl.48 (2003) 235-245. genRefLink(16, ’S0129167X16500701BIB034’, ’10.1080 · Zbl 1037.31004
[35] 35. G. Ren and U. Kähler, Weighted Lipschitz continuity and harmonic Bloch and Besov spaces in the unit real ball, Proc. Edinb. Math. Soc.48 (2005) 743-755. genRefLink(16, ’S0129167X16500701BIB035’, ’10.1017 · Zbl 1148.42308
[36] 36. W. Rudin, Function Theory in the Unit Ball of \(\mathbb{C}\)n, Grundlehren der Mathematischen Wissenschaften, Vol. 241 (Springer, New York, 1980). genRefLink(16, ’S0129167X16500701BIB036’, ’10.1007
[37] 37. A. L. Shields and D. L. Williams, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc.162 (1971) 287-302. genRefLink(128, ’S0129167X16500701BIB037’, ’A1971L327600018’); · Zbl 0227.46034
[38] 38. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, Vol. 32 (Princeton University, Princeton, 1971). · Zbl 0232.42007
[39] 39. S. Stević, On harmonic function spaces, J. Math. Soc. Japan57 (2005) 781-802. genRefLink(16, ’S0129167X16500701BIB039’, ’10.2969
[40] 40. M. Stoll, Harmonic function theory on real hyperbolic space, unpublished manuscript (2000).
[41] 41. K. Stroethoff, Harmonic Bergman spaces, in Holomorphic Spaces, Mathematical Sciences Research Institute Publications, Vol. 33 (Cambridge University, Cambridge, 1998), pp. 51-63. · Zbl 0997.46036
[42] 42. Z. Wu, Operators on harmonic Bergman spaces, Integral Equations Operator Theory24 (1996) 352-371. genRefLink(16, ’S0129167X16500701BIB042’, ’10.1007
[43] 43. R. Yoneda, A characterization of the harmonic Bloch space and the harmonic Besov spaces by an oscillation, Proc. Edinb. Math. Soc.45 (2002) 229-239. genRefLink(16, ’S0129167X16500701BIB043’, ’10.1017
[44] 44. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, Vol. 226 (Springer, New York, 2005). · Zbl 1067.32005
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.