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projections on non-smooth planar domains, J. Geom. Anal., 14, 2004, 1, 63–86 [35] Lanzani, L. and Stein, E. M., Cauchy-type integrals in several complex variables, Bull. Math. Sci., 3, 2013, 2, 241–285 [36] Lanzani, L. and Stein, E. M., Hardy spaces of holomorphic functions for domains in Cn with minimal smoothness, ArXiv e-prints, 2015, jun, http://adsabs.harvard.edu/abs/2015arXiv150603748L [37] Lanzani, L. and Stein, E. M., The Cauchy-Szeg˝o projection for domains in Cn with minimal smoothness, ArXiv e-prints, 2015, jun, http://adsabs.harvard.edu/abs/2015arXiv150603965

. Cruz-Uribe, L. Diening and P. H¨ast¨o, The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14, No 3 (2011), 361-374; DOI: 10.2478/s13540-011-0023-7; http://www.degruyter.com/view/j/fca.2011.14.issue-3/issue-files/fca.2011.14.issue-3.xml. [6] D. Cruz-Uribe SFO and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer Science and Business Media (2013). [7] D. Cruz-Uribe SFO and L. Daniel Wang, Variable Hardy spaces. Indiana Univ. Math. J. 63, No 2 (2014), 447-493. [8] L. Diening, Maximal functions on

/Neumann problems and Hardy classes for the planar conductivity equation, Compl. Var. Elliptic Eq., 59 (4), 504-538, 2014. [6] L. Baratchart, J. Leblond, S. Rigat and E. Russ, Hardy spaces of the conjugate Beltrami equation, J. Funct. Anal., 259 (2), 384-427, 2010. [7] L. Baratchart, E. Pozzi, and E. Russ, Smirnov classes of pseudo-analytic functions, in Preparation, 2017. [8] L. Bers, L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coeffcients and its applications, Conv. Int. EDP, Cremonese, Roma, 111-138, 1954. [9] P. S. Bourdon

1 Introduction This paper aims to study the mapping properties of sublinear operators on weighted Hardy spaces with variable exponents. The weighted Hardy spaces with variable exponents H ω p ⁢ ( ⋅ ) {H^{p(\,\cdot\,)}_{\omega}} was studied in [ 17 ]. They include weighted Hardy spaces [ 28 , 8 , 32 ] and Hardy spaces with variable exponents [ 3 , 26 ]. In [ 17 ], we have presented the atomic decompositions, the Littlewood–Paley characterizations and the maximal function characterizations of H ω p ⁢ ( ⋅ ) {H^{p(\,\cdot\,)}_{\omega}} . In this paper, we study

1 Introduction Fourier multiplier operators play an important role in classical harmonic analysis, wavelet analysis and in the study of partial differential equations such as wave equations, Schrödinger equations and other dispersive equations. The boundedness of the Fourier multiplier operators on various function spaces is important in such a study. The main purpose of this paper is to establish the Hörmander type multiplier theorem to hold on bi-parameter anisotropic Hardy spaces. To state our main results, we first recall some basics about the classical

Advanced Nonlinear Studies 12 (2012), 533–553 Dual Spaces of Weighted Multi-Parameter Hardy Spaces Associated with the Zygmund Dilation ∗ Xiaolong Han, Guozhen Lu†, Yayuan Xiao Department of Mathematics Wayne State University, Detroit, Michigan 48202 e-mail: xlhan@math.wayne.edu, gzlu@math.wayne.edu, and xiao@math.wayne.edu Received 01 February 2012 Communicated by Shair Ahmad Abstract In this paper, we apply the discrete Littlewood-Paley-Stein analysis to prove the duality theorem of weighted multi-parameter Hardy spaces associated with Zygmund dilations, i

Georgian Mathematical Journal Volume 15 (2008), Number 4, 775–783 COMPACT COMPOSITION OPERATORS ON GENERALIZED HARDY SPACES AJAY K. SHARMA Abstract. We investigate compact composition operators acting on gen- eralized Hardy spaces Hω. In fact, we prove that if ω is a differentiable, subharmonic and strictly increasing function defined on [0,∞), then Cϕ is compact on the generalized Hardy spaces if and only if it is compact on the Hardy space H2. 2000 Mathematics Subject Classification: Primary 47B33, 46E10; Se- condary 30D55. Key words and phrases: Generalized

] Tephnadze G., On the maximal operators of Vilenkin-Fejér means on Hardy spaces, Math. Inequal. Appl., 2013, 16(1), 301–312 [25] Tephnadze G., Strong convergence theorems for Walsh-Fejér means, Acta Math. Hungar., 2014, 142(1), 244–259 http://dx.doi.org/10.1007/s10474-013-0361-5 [26] Tephnadze G., A note on the norm convergence of Vilenkin-Fejér means, Georgian Math. J. (in press) [27] Weisz F., Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Math., 1568, Springer, Berlin, 1994 [28] Weisz F., Cesàro summability of one- and two

1 Introduction The real-variable Hardy space H 1 {H^{1}} was introduced by Stein and Weiss [ 19 , 20 ] as a harmonic analytic substitute for the endpoint Lebesgue space ℒ 1 {\mathcal{L}^{1}} . A larger local Hardy space 𝔥 1 {\mathfrak{h}^{1}} , which is better suited for smooth Fourier multipliers, was studied by Goldberg [ 12 ]. The classical setting is based on the Euclidean space ℝ n {\mathbb{R}^{n}} and arises, somewhat implicitly, from its Laplacian Δ = ∂ 2 / ∂ ⁡ x 1 2 + ⋯ + ∂ 2 / ∂ ⁡ x n 2 {\Delta=\partial^{2}/\partial x_{1}^{2}+\cdots+\partial^{2

1 Introduction Since the Hardy spaces were introduced in the one-parameter setting (see, e.g., [ 6 , 7 , 43 ]), these spaces have been studied extensively both in the one-parameter and multi-parameter settings (see, for example, [ 3 , 5 , 1 , 11 , 12 , 14 , 15 , 22 , 31 , 15 , 36 , 38 , 40 , 41 , 43 , 42 ]). It is well known that the classical H p {H^{p}} space is well suited only to the PDEs with constant coefficients, but not stable under multiplication by test functions. Thus it is not suitable for the study of PDEs associated with variable