A class of integral operators from Lebesgue spaces into harmonic Bergman-Besov or weighted Bloch spaces

Year 2021, Volume: 50 Issue: 3, 811 - 820, 07.06.2021

Ömer Faruk Doğan

https://doi.org/10.15672/hujms.768123

Abstract

We consider a class of two-parameter weighted integral operators induced by harmonic Bergman-Besov kernels on the unit ball of $\mathbb{R}^{n}$ and characterize precisely those that are bounded from Lebesgue spaces $L^{p}_{\alpha}$ into harmonic Bergman-Besov spaces $b^{q}_{\beta}$, weighted Bloch spaces $b^{\infty}_{\beta} $ or the space of bounded harmonic functions $h^{\infty}$, allowing the exponents to be different. These operators can be viewed as generalizations of the harmonic Bergman-Besov projections.

Keywords

integral operator , harmonic Bergman-Besov kernel , harmonic Bergman-Besov space , weighted harmonic Bloch space , harmonic Bergman-Besov projection

References

• [1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, 2nd ed., Grad. Texts in Math., 137, Springer, New York, 2001.
• [2] A.E. Djrbashian and F.A. Shamoian, Topics in the Theory of $A^{p}_{\alpha}$ Spaces, Teubner Texts in Mathematics 105, Leipzig, 1988.
• [3] Ö.F. Doğan, Harmonic Besov spaces with small exponents, Complex Var. Elliptic Equ. 65 (6), 1051–1075, 2020.
• [4] Ö.F. Doğan, A Class of Integral Operators Induced by Harmonic Bergman-Besov Kernels on Lebesgue Classes, arXiv:2002.03193v2 [math.CA], 2020.
• [5] Ö.F. Doğan and A.E. Üreyen, Weighted harmonic Bloch spaces on the ball, Complex Anal. Oper. Theory 12 (5), 1143–1177, 2018.
• [6] S. Gergün, H.T. Kaptanoğlu and A.E. Üreyen, Reproducing kernels for harmonic Besov spaces on the ball, C. R. Math. Acad. Sci. Paris 347, 735–738, 2009.
• [7] S. Gergün, H.T. Kaptanoğlu and A.E. Üreyen, Harmonic Besov spaces on the ball, Int. J. Math. 27 (9), 1650070, 59 pp., 2016.
• [8] M. Jevtić and M. Pavlović, Harmonic Bergman functions on the unit ball in $\mathbb R^n$, Acta Math. Hungar. 85, 81–96, 1999.
• [9] M. Jevtić and M. Pavlović, Harmonic Besov spaces on the unit ball in $\mathbb R^n$, Rocky Mountain J. Math. 31, 1305–1316, 2001.
• [10] H.T. Kaptanoğlu and A.E. Üreyen, Singular integral operators with Bergman-Besov kernels on the ball, Integr. Equ. Oper. Theory 91, 30 pp., 2019.
• [11] S. Pérez-Esteva, Duality on vector-valued weighted harmonic Bergman spaces, Studia Math. 118, 37–47, 1996.
• [12] K. Stroethoff, Harmonic Bergman spaces, in Holomorphic Spaces, Mathematical Sciences Research Institute Publications, 33, Cambridge University, Cambridge, 51– 63, 1998