Taiwanese Journal of Mathematics

Variable Anisotropic Hardy Spaces and Their Applications

Jun Liu, Ferenc Weisz, Dachun Yang, and Wen Yuan

Full-text: Open access


Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the variable anisotropic Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$ associated with $A$, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $H_A^{p(\cdot)}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from $H^{p(\cdot)}_A(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from $H^{p(\cdot)}_A(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.

Article information

Taiwanese J. Math., Volume 22, Number 5 (2018), 1173-1216.

Received: 5 August 2017
Accepted: 13 November 2017
First available in Project Euclid: 29 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 42B30: $H^p$-spaces 42B25: Maximal functions, Littlewood-Paley theory 42B08: Summability 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

(variable) Hardy space expansive matrix grand maximal function (finite) atom Littlewood-Paley function Bochner-Riesz means Weierstrass means


Liu, Jun; Weisz, Ferenc; Yang, Dachun; Yuan, Wen. Variable Anisotropic Hardy Spaces and Their Applications. Taiwanese J. Math. 22 (2018), no. 5, 1173--1216. doi:10.11650/tjm/171101. https://projecteuclid.org/euclid.twjm/1511924486

Export citation


  • E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), no. 3, 213–259.
  • ––––, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math. 584 (2005), 117–148.
  • V. Almeida, J. J. Betancor and L. Rodríguez-Mesa, Anisotropic Hardy-Lorentz spaces with variable exponents, Canad. J. Math. 69 (2017), no. 6, 1219–1273.
  • A. Almeida, P. Harjulehto, P. Höstä and T. Lukkari, Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces, Ann. Mat. Pura Appl. (4) 194 (2015), no. 2, 405–424.
  • A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), no. 5, 1628–1655.
  • M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), no. 781, vi+122 pp.
  • M. Bownik, B. Li, D. Yang and Y. Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J. 57 (2008), no. 7, 3065–3100.
  • ––––, Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr. 283 (2010), no. 3, 392–442.
  • D. Breit, L. Diening and S. Schwarzacher, Finite element approximation of the $p(\cdot)$-Laplacian, SIAM J. Numer. Anal. 53 (2015), no. 1, 551–572.
  • P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Volume 1: One-dimensional theory, Pure and Applied Mathematics 40, Academic Press, New York-London, 1971.
  • A.-P. Calderón, An atomic decomposition of distributions in parabolic $H^p$ spaces, Advances in Math. 25 (1977), no. 3, 216–225.
  • A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, Advances in Math. 16 (1975), 1–64.
  • ––––, Parabolic maximal functions associated with a distribution II, Advances in Math. 24 (1977), no. 2, 101–171.
  • Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406.
  • R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, (French) Étude de certaines intégrales singulières, Lecture Notes in Mathematics 242, Springer-Verlag, Berlin-New York, 1971.
  • ––––, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645.
  • D. Cruz-Uribe, The Hardy-Littlewood maximal operator on variable-$L^p$ spaces, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 147–156, Colecc. Abierta 64, Univ. Sevilla Secr. Publ., Seville, 2003.
  • D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.
  • D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  • L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), no. 2, 245–253.
  • L. Diening, P. Harjulehto, P. Hästö and M. R\ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011.
  • L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), no. 6, 1731–1768.
  • L. Diening and S. Schwarzacher, Global gradient estimates for the $p(\cdot)$-Laplacian, Nonlinear Anal. 106 (2014), 70–85.
  • L. Ephremidze, V. Kokilashvili and S. Samko, Fractional, maximal and singular operators in variable exponent Lorentz spaces, Fract. Calc. Appl. Anal. 11 (2008), no. 4, 407–420.
  • X. Fan, J. He, B. Li and D. Yang, Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces, Sci. China Math. 60 (2017), no. 11, 2093–2154.
  • X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.
  • H. G. Feichtinger and F. Weisz, Wiener amalgams and pointwise summability of Fourier transforms and Fourier series, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 3, 509–536.
  • G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.
  • L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ, 2004.
  • L. Grafakos, L. Liu and D. Yang, Maximal function characterizations of Hardy spaces on RD-spaces and their applications, Sci. China Ser. A 51 (2008), no. 12, 2253–2284.
  • Y. Han, D. Müller and D. Yang, Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), no. 13-14, 1505–1537.
  • ––––, A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp.
  • P. Harjulehto, P. Hästö, V. Latvala and O. Toivanen, Critical variable exponent functionals in image restoration, Appl. Math. Lett. 26 (2013), no. 1, 56–60.
  • Y. Jiao, Y. Zuo, D. Zhou and L. Wu, Variable Hardy-Lorentz spaces $H^{p(\cdot),q}(\mathbb{R}^{n})$, Submitted.
  • H. Kempka and J. Vybíral, Lorentz spaces with variable exponents, Math. Nachr. 287 (2014), no. 8-9, 938–954.
  • O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41 (116) (1991), no. 4, 592–618.
  • L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), no. 1, 115–150.
  • B. Li, M. Bownik and D. Yang, Littlewood-Paley characterization and duality of weighted anisotropic product Hardy spaces, J. Funct. Anal. 266 (2014), no. 5, 2611–2661.
  • B. D. Li, X. Y. Fan, Z. W. Fu and D. C. Yang, Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 11, 1391–1414.
  • B. Li, X. Fan and D. Yang, Littlewood-Paley characterizations of anisotropic Hardy spaces of Musielak-Orlicz type, Taiwanese J. Math. 19 (2015), no. 1, 279–314.
  • B. Li, D. Yang and W. Yuan, Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal 2014 (2014), Article ID 306214, 19 pages.
  • Y. Liang, J. Huang and D. Yang, New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (2012), no. 1, 413–428.
  • Y. Liang, Y. Sawano, T. Ullrich, D. Yang and W. Yuan, New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets, J. Fourier Anal. Appl. 18 (2012), no. 5, 1067–1111.
  • J. Liu, F. Weisz, D. Yang and W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, Submitted.
  • J. Liu, D. Yang and W. Yuan, Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math. 59 (2016), no. 9, 1669–1720.
  • ––––, Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl. 456 (2017), no. 1, 356–393.
  • ––––, Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces, Acta Math. Sci. Ser. B Engl. Ed. (to appear) or arXiv:1601.05242.
  • ––––, Littlewood-Paley characterizations of weighted anisotropic Triebel-Lizorkin spaces via averages on balls, Submitted.
  • S. Z. Lu, Four Lectures on Real $H^p$ Spaces, World Scientific, River Edge, NJ, 1995.
  • S. Meda, P. Sjögren and M. Vallarino, On the $H^1$-$L^1$ boundedness of operators, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2921–2931.
  • J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics 1034, Springer-Verlag, Berlin, 1983.
  • E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  • H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950.
  • ––––, Topology of Linear Topological Spaces, Maruzen, Tokyo, 1951.
  • T. Noi and Y. Sawano, Complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents, J. Math. Anal. Appl. 387 (2012), no. 2, 676–690.
  • W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), no. 1, 200–211.
  • M. R\ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin, 2000.
  • S. Sato, Estimates for singular integrals on homogeneous groups, J. Math. Anal. Appl. 400 (2013), no. 2, 311–330.
  • ––––, Characterization of parabolic Hardy spaces by Littlewood-Paley functions, arXiv:1607.03645.
  • Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148.
  • H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, A Wiley-Interscience Publication, John Wiley & Sons, Chichester, 1987.
  • E. M. Stein, M. H. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), Rend. Circ. Mat. Palermo (2) 1981, suppl. 1, 81–97.
  • L. Tang, $L^{p(\cdot),\lambda(\cdot)}$ regularity for fully nonlinear elliptic equations, Nonlinear Anal. 149 (2017), 117–129.
  • J. Tiirola, Image decompositions using spaces of variable smoothness and integrability, SIAM J. Imaging Sci. 7 (2014), no. 3, 1558–1587.
  • H. Triebel, Theory of Function Spaces III, Monographs in Mathematics 100, Birkhäuser Verlag, Basel, 2006.
  • ––––, Tempered Homogeneous Function Spaces, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015.
  • R. M. Trigub and E. S. Bellinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers, Dordrecht, 2004.
  • T. Ullrich, Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits, J. Funct. Spaces Appl. 2012, Art. ID 163213, 47 pp.
  • J. Vybíral, Sobolev and Jawerth embeddings for spaces with variable smoothness and integrability, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 529–544.
  • F. Weisz, Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and its Applications 541, Kluwer Academic Publishers, Dordrecht, 2002.
  • ––––, Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory 7 (2012), 1–179.
  • ––––, Convergence and Summability of Fourier Transforms and Hardy Spaces, Birkhäuser, Basel, 2017.
  • J. Xu, Variable Besov and Triebel-Lizorkin spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 2, 511–522.
  • X. Yan, D. Yang, W. Yuan and C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), no. 10, 2822–2887.
  • D. Yang, Y. Liang and L. D. Ky, Real-variable Theory of Musielak-Orlicz Hardy Spaces, Lecture Notes in Mathematics 2182, Springer, Cham, 2017.
  • D. Yang, D. Yang and G. Hu, The Hardy Space $H^1$ with Non-doubling Measures and Their Applications, Lecture Notes in Mathematics 2084, Springer, Cham, 2013.
  • D. Yang, W. Yuan and C. Zhuo, Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces, Rev. Mat. Complut. 27 (2014), no. 1, 93–157.
  • D. Yang and Y. Zhou, Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms, J. Math. Anal. Appl. 339 (2008), no. 1, 622–635.
  • ––––, A boundedness criterion via atoms for linear operators in Hardy spaces, Constr. Approx. 29 (2009), no. 2, 207–218.
  • ––––, New properties of Besov and Triebel-Lizorkin spaces on RD-spaces, Manuscripta Math. 134 (2011), no. 1-2, 59–90.
  • D. Yang, C. Zhuo and E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), no. 2, 245–270.
  • D. Yang, C. Zhuo and W. Yuan, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal. 269 (2015), no. 6, 1840–1898.
  • C. Zhuo, Y. Sawano and D. Yang, Hardy spaces with variable exponents on RD-spaces and applications, Dissertationes Math. (Rozprawy Mat.) 520 (2016), 1–74.
  • C. Zhuo, D. Yang and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1541–1577.
  • C. Zhuo, D. Yang and W. Yuan, Interpolation between $H^{p(\cdot)}(\mathbb{R}^{n})$ and $L^{\infty}(\mathbb{R}^{n})$: real method, J. Geom. Anal. (2017) or arXiv:1703.05527.