Harmonic extension of Q-type space related to logarithmic functions

Abstract

This paper studies a class of Q-type spaces Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)related to logarithmic functions. We first investigate some basic properties of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). Further, by the aid of Poisson integral and harmonic function spaces H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right), the harmonic extension of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)and the boundary value problem of H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right)are obtained.

Keywords

Q type space / Poisson integral / harmonic extension

References

[1] Aulaskari R, Xiao J, Zhao R H. On subspaces and subsets of BMOA and UBC. Analysis 1995; 15(2): 101–121

[2] BaoG L. Characterizations of Q_k spaces of several real variables. Ph D Thesis, Shantou: Shantou University, 2014 (in Chinese)

[3] Cobos F, Dominguez Ó, Triebel H. Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-Groups. J Funct Anal 2016; 270(12): 4386–4425

[4] Dafni G, Xiao J. Some new tent spaces and duality theorems for fractional Carleson measures and Qα(Rn). J Funct Anal 2004; 208(2): 377–422

[5] Essén M, Janson S, Peng L Z, Xiao J. Q spaces of several real variables. Indiana Univ Math J 2000; 49(2): 575–615

[6] Koskela P, Xiao J, Zhang Y R-Y, Zhou Y. A quasiconformal composition problem for the Q-spaces. J Eur Math Soc (JEMS) 2017; 19(4): 1159–1187

[7] KufnerAPerssonL-E. Weighted Inequalities of Hardy Type. River Edge, NJ: World Scientific, 2003

[8] Li P T, Yang Q X, Zhao K. Regular wavelets and Triebel-Lizorkin type oscillation spaces. Math Methods Appl Sci 2017; 40(18): 6684–6701

[9] Li P T, Zhai Z C. Well-posedness and regularity of generalized Navier-Stokes equations in some critical Q-spaces. J Funct Anal 2009; 259(10): 2457–2519

[10] Li P T, Zhai Z C. Riesz transforms on Q-type spaces with application to quasi-geostrophic equation. Taiwanese J Math 2012; 16(6): 2107–2132

[11] Liu L G, Wu S Q, Xiao J, Yuan W. The logarithmic Sobolev capacity. Adv Math 2021; 392: 107993

[12] Stegenga D A. Multipliers of the Dirichlet space. Illinois J Math 1980; 24(1): 113–139

[13] TrefethenL NBauD. III, Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997

[14] Wu Z J, Xie C P. Q spaces and Morrey spaces. J Funct Anal 2003; 201(1): 282–297

[15] Xiao J. Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn Partial Differ Equ 2007; 4(3): 227–245

[16] Yang Q X, Qian T, Li P T. Fefferman-Stein decomposition for Q-spaces and micro-local quantities. Nonlinear Anal 2016; 145: 24–48

RIGHTS & PERMISSIONS

Higher Education Press 2024