Discrete Paraproduct Operators on Variable Hardy Spaces

Tan, Jian

Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 304-317

Résumé

Let $p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators $\unicode[STIX]{x1D70B}_{b}$ on variable Hardy spaces $H^{p(\cdot )}(\mathbb{R}^{n})$, where $b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators $T$ are bounded on $H^{p(\cdot )}(\mathbb{R}^{n})$ if and only if $T^{\ast }1=0$, where $\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$ and $\unicode[STIX]{x1D716}$ is the regular exponent of kernel of $T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

Détail

DOI : 10.4153/S0008439519000298

Mots-clés : variable Hardy space, singular integral, paraproduct operator, discrete Littlewood–Paley–Stein theory

Bibliographie

[1] Bony, J.-M., Interaction des singularités pour les équations de Klein-Gordon non linéaires. In: Goulaouic–Meyer–Schwartz seminar, 1983-1984, Exp. No. 10. École Polytech., Palaiseau. Google Scholar

[2] Coifman, R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes. (Lecture Notes in Mathematics, 242), Springer-Verlag, Berlin, 1971.10.1007/BFb0058946 Google Scholar

[3] Cruz-Uribe, D. and Fiorenza, A., Variable Lebesgue spaces. Birkhäuser/Springer Heidelberg, 2013. https://doi.org/10.1007/978-3-0348-0548-3 Google Scholar

[4] Cruz-Uribe, D., Fiorenza, A., Martell, J., and Pérez, C., The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 31(2006), 239–264. Google Scholar

[5] Cruz-Uribe, D., Moen, K., and Nguyen, H. V., The boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces. Publ. Mat. 63(2019), 679–713. https://doi.org/10.5565/PUBLMAT6321908 Google Scholar

[6] Cruz-Uribe, D., Moen, K., and Nguyen, H. V., A new approach to norm inequalities on weighted and variable Hardy spaces. Google Scholar

[7] Cruz-Uribe, D. and Wang, L., Variable Hardy spaces. Indiana Univ. Math. J. 63(2014), 447–493. https://doi.org/10.1512/iumj.2014.63.5232 Google Scholar

[8] Deng, D.-G. and Han, Y.-S., The theory of H p spaces. Peking University Press, Beijing, 1992. Google Scholar

[9] Deng, D.-G. and Han, Y.-S., Harmonic analysis on spaces of homogeneous type. (Lecture Notes in Mathematics, 1966), Springer-Verlag, Berlin, 2009.10.1007/978-3-540-88745-4 Google Scholar

[10] Diening, L., Harjulehto, P., Hästö, P., and Růžička, M., Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8 Google Scholar

[11] Fefferman, C. and Stein, E., H p spaces of several variables. Acta Math. 129(1972), no. 3–4, 137–193. https://doi.org/10.1007/BF02392215 Google Scholar

[12] Frazier, M. and Jawerth, B., Decomposition of Besov spaces. Indiana Univ. Math. J. 34(1985), no. 4, 777–799. https://doi.org/10.1512/iumj.1985.34.34041 Google Scholar

[13] Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1990), no. 1, 34–170. https://doi.org/10.1016/0022-1236(90)90137-A Google Scholar

[14] Han, Y.-S., Calderón-type reproducing formula and the T theorem. Rev. Mat. Iberoamericana 10(1994), no. 1, 51–91. https://doi.org/10.4171/RMI/145 Google Scholar

[15] Han, Y.-S., Müller, D., and Yang, D.-C., A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal.(2008), Art. ID 893409. https://doi.org/10.1155/2008/893409 Google Scholar

[16] Han, Y.-S., Müller, D., and Yang, D.-C., Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279(2006), no. 13-14, 1505–1537. https://doi.org/10.1002/mana.200610435 Google Scholar

[17] Han, Y.-S. and Sawyer, E., Para-accretive functions, the weak boundedness property and the Tb theorem. Rev. Mat. Iberoamericana 6(1990), 17–41. https://doi.org/10.4171/RMI/93 Google Scholar

[18] Han, Y.-S. and Sawyer, E. T., Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Mem. Amer. Math. Soc. 110(1994), no. 530. https://doi.org/10.1090/memo/0530 Google Scholar

[19] Hung, C.-C. and Lee, M.-Y., The boundedness of Calderón-Zygmund operators by wavelet characterization. Acta Math. Sin. (Engl. Ser.) 28(2012), no. 6, 1237–1248. https://doi.org/10.1007/s10114-011-0065-0 Google Scholar

[20] Kováčik, O. and Rákosník, J., On spaces L p (x) and W k, p (x). Czechoslovak Math. J. 41(1991), 592–618. Google Scholar

[21] Meyer, Y., Wavelets and operators. (Cambridge Studies in Advanced Mathematics, 37), Cambridge University Press, Cambridge, 1992. Google Scholar

[22] Nakai, E. and Sawano, Y., Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262(2012), 3665–3748. https://doi.org/10.1016/j.jfa.2012.01.004 Google Scholar

[23] Pick, L. and Růžičkap, M., An example of a space L p (x) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math. 19(2001), 369–371. https://doi.org/10.1016/S0723-0869(01)80023-2 Google Scholar

[24] Sawano, Y., Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equations Operator Theory 77(2013), 123–148. https://doi.org/10.1007/s00020-013-2073-1 Google Scholar

[25] Stein, E. M. and Weiss, G., On the theory of harmonic functions of several variables, I. The theory of H p-spaces. Acta Math. 103(1960), 25–62. https://doi.org/10.1007/BF02546524 Google Scholar

[26] Tan, J., Atomic decomposition of variable Hardy spaces via Littlewood-Paley-Stein theory. Ann. Funct. Anal. 9(2018), no. 1, 87–100. https://doi.org/10.1215/20088752-2017-0026 Google Scholar

[27] Tan, J., Bilinear Calderón-Zygmund operators on products of variable Hardy spaces. Forum Math. 31(2019), no. 1, 187–198. https://doi.org/10.1515/forum-2018-0082 Google Scholar

[28] Tan, J., Atomic decompositions of localized Hardy spaces with variable exponents and applications. J. Geom. Anal. 29(2019), no. 1, 799–827. https://doi.org/10.1007/s12220-018-0019-1 Google Scholar

[29] Tan, J., Boundedness of maximal operator for multilinear Calderón-Zygmund operators on products of variable Hardy spaces. Kyoto J. Math.(2018), to appear. Google Scholar

[30] Tan, J., Boundedness of multilinear fractional type operators on Hardy spaces with variable exponents, preprint. Google Scholar

[31] Yang, D.-C., Zhuo, C.-Q., and Nakai, E., Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev. Mat. Complut. 29(2016), no. 2, 245–270. https://doi.org/10.1007/s13163-016-0188-z Google Scholar

[32] Zhuo, C.-Q., Sawano, Y., and Yang, D.-C., Hardy spaces with variable exponents on RD-spaces and applications. Dissertationes Math. 520(2016). https://doi.org/10.4064/dm744-9-2015 Google Scholar