Abstract
Let \(p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]\) be a variable exponent function satisfying the globally log-Hölder continuous condition, \(q\in (0,\infty ]\) and A be a general expansive matrix on \({\mathbb {R}}^n\). Let \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) by means of the Littlewood–Paley g-function or the Littlewood–Paley \(g_\lambda ^*\)-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space \(L^{p(\cdot ),q}({\mathbb {R}}^n)\). Moreover, the finite atomic characterization of \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) to \(L^{p(\cdot ),q}({\mathbb {R}}^n)\) 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.
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Abu-Shammala, W., Torchinsky, A.: The Hardy-Lorentz spaces \(H^{p, q}({\mathbb{R}}^n)\). Stud. Math. 182, 283–294 (2007)
Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)
Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)
Almeida, A., Caetano, A.M.: Generalized Hardy spaces. Acta Math. Sin. (Engl. Ser.) 26, 1673–1692 (2010)
Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258, 1628–1655 (2010)
Almeida, A., Harjulehto, P., Höstä, P., Lukkari, T.: Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces. Ann. Mat. Pura Appl. (4) 194, 405–424 (2015)
Almeida, V., Betancor, J.J., Rodríguez-Mesa, L.: Anisotropic Hardy-Lorentz spaces with variable exponents. Can. J. Math. 69, 1219–1273 (2017)
Álvarez, J.: \(H^p\) and weak \(H^p\) continuity of Calderón-Zygmund type operators. In: Fourier Analysis, Lecture Notes in Pure and Appl. Math., vol. 157, pp. 17-34, Dekker, New York (1994)
Aoki, T.: Locally bounded linear topological spaces. Proc. Imp. Acad. Tokyo 18, 588–594 (1942)
Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer, Berlin (1976)
Bownik, M.: Anisotropic Hardy Spaces and Wavelets, Mem. Amer. Math. Soc. 164 , no. 781 (2003)
Bownik, M., Ho, K.-P.: Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans. Am. Math. Soc. 358, 1469–1510 (2006)
Bownik, M., Li, B., Yang, D., Zhou, Y.: Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators. Indiana Univ. Math. J. 57, 3065–3100 (2008)
Breit, D., Diening, L., Schwarzacher, S.: Finite element approximation of the \(p(\cdot )\)-Laplacian. SIAM J. Numer. Anal. 53, 551–572 (2015)
Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation, Volume 1: One-Dimensional Theory, Pure and Applied Mathematics, vol. 40. Academic Press, New York-London (1971)
Calderón, A.-P.: Intermediate spaces and interpolation. The complex method. Studia Math. 24, 113–190 (1964)
Calderón, A.-P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. 53, 1092–1099 (1965)
Calderón, A.-P.: An atomic decomposition of distributions in parabolic \(H^p\) spaces. Adv. Math. 25, 216–225 (1977)
Calderón, A.-P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. Adv. Math. 16, 1–64 (1975)
Calderón, A.-P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. II. Adv. Math. 24, 101–171 (1977)
Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)
Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)
Coifman, R.R., Weiss, G.: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, (French) Étude de certaines intégrales singulières. Lecture Notes in Mathematics, vol. 242, Springer, Berlin (1971)
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg (2013)
Cruz-Uribe, D., Wang, L.-A.D.: Variable Hardy spaces. Indiana Univ. Math. J. 63, 447–493 (2014)
Cruz-Uribe, D., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)
Dekel, S., Han, Y., Petrushev, P.: Anisotropic meshless frames on \({\mathbb{R}}^n\). J. Fourier Anal. Appl. 15, 634–662 (2009)
Diening, L., Schwarzacher, S.: Global gradient estimates for the \(p(\cdot )\)-Laplacian. Nonlinear Anal. 106, 70–85 (2014)
Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731–768 (2009)
Diening, L., Harjulehto, P., Hästö, P., R\(\mathring{\rm u}\)žička, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. Springer, Heidelberg (2011)
Ding, Y., Sato, S.: Littlewood-Paley functions on homogeneous groups. Forum Math. 28, 43–55 (2016)
Fan, X., He, J., Li, B., Yang, D.: Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces. Sci. China Math. 60, 2093–2154 (2017)
Fefferman, R., Soria, F.: The spaces weak \(H^1\). Stud. Math. 85, 1–16 (1987)
Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)
Fefferman, C., Rivière, N.M., Sagher, Y.: Interpolation between \(H^p\) spaces: the real method. Trans. Am. Math. Soc. 191, 75–81 (1974)
Feichtinger, H.G., Weisz, F.: Wiener amalgams and pointwise summability of Fourier transforms and Fourier series. Math. Proc. Cambridge Philos. Soc. 140, 509–536 (2006)
Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups, Mathematical Notes, vol. 28. Princeton University Press, University of Tokyo Press, Princeton, NJ (1982)
Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and The Study of Function Spaces. In: CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1991)
Gát, G.: Pointwise convergence of cone-like restricted two-dimensional \((C,1)\) means of trigonometric Fourier series. J. Approx. Theory 149, 74–102 (2007)
Gát, G.: Almost everywhere convergence of sequences of Cesàro and Riesz means of integrable functions with respect to the multidimensional Walsh system. Acta Math. Sin. (Engl. Ser.) 30, 311–322 (2014)
Goginava, U.: Marcinkiewicz-Fejer means of \(d\)-dimensional Walsh-Fourier series. J. Math. Anal. Appl. 307, 206–218 (2005)
Goginava, U.: Almost everywhere convergence of \((C,\alpha )\)-means of cubical partial sums of \(d\)-dimensional Walsh-Fourier series. J. Approx. Theory 141, 8–28 (2006)
Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, vol. 249, 3rd edn. Springer, New York (2014)
Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 3rd edn. Springer, New York (2014)
Grafakos, L., Liu, L., Yang, D.: Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci. China Ser. A 51, 2253–2284 (2008)
Han, Y., Müller, D., Yang, D.: Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279, 1505–1537 (2006)
Han, Y., Müller, D., Yang, D.: A Theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal., Art. ID 893409 (2008)
Harjulehto, P., Hästö, P., Latvala, V., Toivanen, O.: Critical variable exponent functionals in image restoration. Appl. Math. Lett. 26, 56–60 (2013)
Hunt, R.A.: On the convergence of Fourier series. In: Orthogonal Expansions and Their Continuous Analogues, Proc. Conf. Edwardsville, Ill., 1967, pp. 235–255. Illinois Univ. Press, Carbondale (1968)
Ioku, N., Ishige, K., Yanagida, E.: Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups. J. Math. Pures Appl. (9) 103, 900–923 (2015)
Jakab, T., Mitrea, M.: Parabolic initial boundary value problems in nonsmooth cylinders with data in anisotropic Besov spaces. Math. Res. Lett. 13, 825–831 (2006)
Jiao, Y., Zuo, Y., Zhou, D., Wu, L.: Variable Hardy-Lorentz spaces \(H^{p(\cdot ),q}(\mathbb{R}^{n})\) (submitted)
Kempka, H., Vybíral, J.: Lorentz spaces with variable exponents. Math. Nachr. 287, 938–954 (2014)
Ky, L.D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equ. Oper. Theory 78, 115–150 (2014)
Li, B., Bownik, M., Yang, D., Yuan, W.: Duality of weighted anisotropic Besov and Triebel-Lizorkin spaces. Positivity 16, 213–244 (2012)
Li, B., Bownik, M., Yang, D.: Littlewood-Paley characterization and duality of weighted anisotropic product Hardy spaces. J. Funct. Anal. 266, 2611–2661 (2014)
Liang, Y., Huang, J., Yang, D.: New real-variable characterizations of Musielak-Orlicz Hardy spaces. J. Math. Anal. Appl. 395, 413–428 (2012)
Liang, Y., Sawano, Y., Ullrich, T., Yang, D., Yuan, W.: New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets. J. Fourier Anal. Appl. 18, 1067–1111 (2012)
Liang, Y., Yang, D., Jiang, R.: Weak Musielak-Orlicz Hardy spaces and applications. Math. Nachr. 289, 634–677 (2016)
Lions, J.-L., Peetre, J.: Sur une classe despaces dinterpolation (French). Inst. Hautes tudes Sci. Publ. Math. 19, 5–68 (1964)
Littlewood, J.E., Paley, R.E.A.C.: Theorems on Fourier series and power series. J. Lond. Math. Soc. 6, 230–233 (1931)
Liu, H.: The weak \(H^p\) spaces on homogeneous groups. In: Harmonic Analysis (Tianjin, 1988), Lecture Notes in Math., vol. 1494, pp. 113–118. Springer, Berlin (1991)
Liu, J., Yang, D., Yuan, W.: Anisotropic Hardy-Lorentz spaces and their applications. Sci. China Math. 59, 1669–1720 (2016)
Liu, J., Yang, D., Yuan, W.: Anisotropic variable Hardy-Lorentz spaces and their real interpolation. J. Math. Anal. Appl. 456, 356–393 (2017)
Liu, J., Yang, D., Yuan, W.: Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces. Acta Math. Sci. B 38, 1–33 (2018)
Liu, J., Yang, D., Yuan, W.: Littlewood-Paley characterizations of weighted anisotropic Triebel-Lizorkin spaces via averages on balls (submitted)
Lorentz, G.G.: Some new functional spaces. Ann. Math. (2) 51, 37–55 (1950)
Lorentz, G.G.: On the theory of spaces \(\Lambda \). Pac. J. Math. 1, 411–429 (1951)
Lu, S.-Z.: Four Lectures on Real \(H^p\) Spaces. World Scientific Publishing Co., Inc, River Edge, NJ (1995)
Lu, S.-Z., Yan, D.: Bochner-Riesz Means on Euclidean Spaces. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2013)
Meda, S., Sjögren, P., Vallarino, M.: On the \(H^1\)-\(L^1\) boundedness of operators. Proc. Am. Math. Soc. 136, 2921–2931 (2008)
Merucci, C.: Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces. In: Interpolation Spaces and Allied Topics in Analysis (Lund, 1983). Lecture Notes in Math, vol. 1070, pp. 183–201. Springer, Berlin (1984)
Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math, vol. 1034. Springer, Berlin (1983)
Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)
Nakano, H.: Modulared Semi-ordered Linear Spaces. Maruzen Co., Ltd, Tokyo (1950)
Nakano, H.: Topology of Linear Topological Spaces. Maruzen Co., Ltd, Tokyo (1951)
Noi, T.: Trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents. Rev. Mat. Complut. 29, 341–404 (2016)
Noi, T., Sawano, Y.: Complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents. J. Math. Anal. Appl. 387, 676–690 (2012)
Oberlin, R., Seeger, A., Tao, T., Thiele, C., Wright, J.: A variation norm Carleson theorem. J. Eur. Math. Soc. (JEMS) 14, 421–464 (2012)
Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3, 200–212 (1931)
Orlicz, W.: Über eine gewisse Klasse von Räumen vom Typus B. Bull. Int. Acad. Pol. A 8, 207–220 (1932)
Parilov, D.: Two theorems on the Hardy-Lorentz classes \(H^{1,q}\), (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 327 (2005), Issled. po Linein. Oper. i Teor. Funkts. 33, 150–167, 238; translation in J. Math. Sci. (N. Y.) 139, 6447–6456 (2006)
Peetre, J.: Nouvelles propriétés d’espaces d’interpolation. C. R. Acad. Sci. Paris 256, 1424–1426 (1963)
Persson, L.-E., Tephnadze, G., Wall, P.: Maximal operators of Vilenkin-Nörlund means. J. Fourier Anal. Appl. 21, 76–94 (2015)
Phuc, N.C.: The Navier-Stokes equations in nonendpoint borderline Lorentz spaces. J. Math. Fluid Mech. 17, 741–760 (2015)
R\(\mathring{\rm u}\)žička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)
Rolewicz, S.: On a certain class of linear metric spaces. Bull. Acad. Polon. Sci. Cl. Trois. 5, 471–473 (1957)
Sato, S.: Characterization of parabolic Hardy spaces by Littlewood-Paley functions. arXiv: 1607.03645
Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operator. Integral Equ. Oper. Theory 77, 123–148 (2013)
Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)
Simon, P.: Cesaro summability with respect to two-parameter Walsh systems. Monatsh. Math. 131, 321–334 (2000)
Simon, P.: \((C,\alpha )\) summability of Walsh-Kaczmarz-Fourier series. J. Approx. Theory 127, 39–60 (2004)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, with the assistance of Timothy S. Murphy, Princeton Mathematical Series, Monographs in Harmonic Analysis, III, vol. 43. Princeton University Press, Princeton, NJ (1993)
Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables, I: the theory of \(H^p\) spaces. Acta Math. 103, 25–62 (1960)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton, NJ (1971)
Stein, E.M., Taibleson, M.H., Weiss, G.: Weak type estimates for maximal operators on certain \(H^p\) classes. In: Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), Rend. Circ. Mat. Palermo (2), suppl. 1, 81–97 (1981)
Tang, L.: \(L^{p(\cdot ),\lambda (\cdot )}\) regularity for fully nonlinear elliptic equations. Nonlinear Anal. 149, 117–129 (2017)
Tiirola, J.: Image decompositions using spaces of variable smoothness and integrability. SIAM J. Imaging Sci. 7, 1558–1587 (2014)
Triebel, H.: Theory of Function Spaces. II. Birkhäuser Verlag, Basel (1992)
Triebel, H.: Theory of Function Spaces. III. Birkhäuser Verlag, Basel (2006)
Triebel, H.: Tempered Homogeneous Function Spaces. European Mathematical Society (EMS), Zürich, EMS Series of Lectures in Mathematics (2015)
Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers, Dordrecht (2004)
Ullrich, T.: Continuous characterization of Besov-Lizorkin-Triebel space and new interpretations as coorbits. J. Funct. Space Appl., Art. ID 163213, 47 pp (2012)
Vybíral, J.: Sobolev and Jawerth embeddings for spaces with variable smoothness and integrability. Ann. Acad. Sci. Fenn. Math. 34, 529–544 (2009)
Weisz, F.: Summability of Multi-dimensional Fourier Series and Hardy Spaces. Mathematics and Its Applications, vol. 541. Kluwer Academic Publishers, Dordrecht (2002)
Weisz, F.: Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory 7, 1–179 (2012)
Weisz, F.: Lebesgue points of two-dimensional Fourier transforms and strong summability. J. Fourier Anal. Appl. 21, 885–914 (2015)
Weisz, F.: Convergence and Summability of Fourier Transforms and Hardy Spaces. Birkhäuser, Basel (2017)
Xu, J.: Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn. Math. 33, 511–522 (2008)
Xu, J.: The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces. Integr. Transform Spec. Funct. 19, 599–605 (2008)
Yan, X., Yang, D., Yuan, Y., Zhuo, C.: Variable weak Hardy spaces and thier applications. J. Funct. Anal. 271, 2822–2887 (2016)
Yang, D., Zhou, Y.: Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms. J. Math. Anal. Appl. 339, 622–635 (2008)
Yang, D., Zhou, Y.: A boundedness criterion via atoms for linear operators in Hardy spaces. Constr. Approx. 29, 207–218 (2009)
Yang, D., Zhou, Y.: New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscripta Math. 134, 59–90 (2011)
Yang, D., Yang, Do, Hu, G.: The Hardy Space \(H^1\) with Non-doubling Measures and Their Applications. Lecture Notes in Mathematics, vol. 2084. Springer, Cham (2013)
Yang, D., Zhuo, C., Yuan, W.: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. 269, 1840–1898 (2015)
Yang, D., Zhuo, C., Nakai, E.: Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev. Mat. Complut. 29, 245–270 (2016)
Yang, D., Liang, Y., Ky, L.D.: Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol. 2182. Springer, Cham (2017)
Zhuo, C., Yang, D., Liang, Y.: Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 39, 1541–1577 (2016)
Zhuo, C., Yang, D., Yuan, W.: Interpolation between \(H^{p(\cdot )}({\mathbb{R}^n) and L^{\infty }(\mathbb{R}}^n)\): real method. J. Geom. Anal. (2017). arXiv: 1703.05527
Acknowledgements
Jun Liu would like to express his deep thanks to Professor Ciqiang Zhuo for several useful conversations on Lemma 3.5 which is an important tool of this article and is of independent interests. Wen Yuan would like to thank Professor Marcin Bownik for a helpful discussion on the subject of this article. The authors would also like to thank both referees for their several stimulating remarks which do improve the presentation of this article.
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Communicated by Hans G. Feichtinger.
This project is supported by the National Natural Science Foundation of China (Grant Nos. 11571039, 11726621, 11761131002 and 11471042) and also by the Hungarian Scientific Research Funds (OTKA) No. K115804.
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Liu, J., Weisz, F., Yang, D. et al. Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications. J Fourier Anal Appl 25, 874–922 (2019). https://doi.org/10.1007/s00041-018-9609-3
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DOI: https://doi.org/10.1007/s00041-018-9609-3
Keywords
- Variable exponent
- (Hardy-)Lorentz space
- Expansive matrix
- Finite atom
- Littlewood–Paley function
- Bochner–Riesz means
- Weierstrass means