Abstract
In this paper, we introduce the Carleson measure spaces with variable exponents \(CMO^{p(\cdot )}\). By using discrete Littlewood–Paley–Stein analysis as well as Frazier and Jawerth’s \(\varphi -\)transform in the variable exponent settings, we show that the dual spaces of the variable Hardy spaces \(H^{p(\cdot )}\) are \(CMO^{p(\cdot )}\). As applications, we obtain that Carleson measure spaces with variable exponents \(CMO^{p(\cdot )}\), Campanato space with variable exponent \({\mathfrak {L}}_{q,p(\cdot ),d}\) and Hölder–Zygmund spaces with variable exponents \(\mathcal {\dot{H}}_d^{p(\cdot )}\) coincide as sets and the corresponding norms are equivalent. Via using an argument of weak density property, we also prove that Calderón–Zygmund singular integral operators are bounded on \(CMO^{p(\cdot )}\).
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References
Coifman, R.: A real variable characterization of \(H^p\). Studia Math. 51, 269–274 (1974)
Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Birkhäuser, Basel (2013)
Cruz-Uribe, D., Wang, L.: Variable Hardy spaces. Indiana Univ. Math. J. 63(2), 447–493 (2014)
Cruz-Uribe, D., Fiorenza, A., Martell, J., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)
Deng, D., Han, Y.-S.: Theory of \(H^p\) Spaces. Peking University Press, Beijing (1992)
Deng, D., Han, Y.-S.: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol. 1966. Springer, Berlin (2009)
Diening, L., Harjulehto, P., Hästö, P., R\(\mathring{u}\)z̆ic̆ka, M.: Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-18363-8
Diening, L.: Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129(8), 657–700 (2005)
Fefferman, C., Stein, E.: \(H^p\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)
Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)
Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 3rd edn. Springer, New York (2014)
Han, Y.-C., Han, Y.-S.: Boundedness of composition operators associated with mixed homogeneities on Lipschitz spaces. Math. Res. Lett. 23(5), 1387–1403 (2016)
Han, Y.-S., Li, J., Lu, G.: Duality of multiparameter Hardy spaces \(H^p\) on spaces of homogeneous type. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9(4), 645–685 (2010)
Izuki, M.: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59, 199–213 (2010)
Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)
Lee, M.-Y., Lin, C.-C.: Carleson measure spaces associated to para-accretive functions. Commun. Contemp. Math 14(1), 1250002 (2012). 19 pp
Lee, M.-Y., Lin, C.-C., Lin, Y.-C.: A wavelet characterization for the dual of weighted Hardy spaces. Proc. Am. Math. Soc. 137, 4219–4225 (2009)
Li, J., Ward, L.: Singular integrals on Carleson measure spaces \(CMO^p\) on product spaces of homogeneous type. Proc. Am. Math. Soc. 141(8), 2767–2782 (2013)
Lin, C.-C., Wang, K.: Generalized Carleson measure spaces and their applications. Abstr. Appl. Anal. Article ID 879073, pp. 26 (2012)
Lin, C.-C.: Boundedness of Monge–Ampére singular integral operators acting on Hardy spaces and their duals. Trans. Am. Math. Soc. 368(5), 3075–3104 (2016)
Lin, C.-C., Wang, K.: Calderón–Zygmund operators acting on generalized Carleson measure spaces. Studia Math. 211(3), 231–240 (2012)
Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)
Tan, J.: Discrete para-product operators on variable Hardy spaces. Can. Math. Bull. https://doi.org/10.4153/S0008439519000298. arXiv:1903.10094, (2019)
Tan, J.: Atomic decomposition of variable Hardy spaces via Littlewood–Paley–Stein theory. Ann. Funct. Anal. 9(1), 87–100 (2018)
Tan, J.: Atomic decompositions of localized Hardy spaces with variable exponents and applications. J. Geom. Anal. 29(1), 799–827 (2019)
Tan, J., Han, Y.-C.: Inhomogeneous multi-parameter Lipschitz spaces associated with different homogeneities and their applications. Filomat 32(9), 3397–3408 (2018)
Tan, J., Zhao, J.: Inhomogeneous Lipschitz spaces of variable order and their applications. Ann. Funct. Anal. 9(1), 72–86 (2018)
Tan, J., Liu, Z., Zhao, J.: On some multilinear commutators in variable Lebesgue spaces. J. Math. Inequal. 11(3), 715–734 (2017)
Yang, D., Zhuo, C., Yuan, W.: Triebel–Lizorkin type spaces with variable exponents. Banach J. Math. Anal. 9(4), 146–202 (2015)
Yang, D., Zhuo, C., Liang, Y.: Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 39(4), 1541–1577 (2016)
Acknowledgements
The Project is sponsored by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20180734), Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 18KJB110022) and Nanjing University of Posts and Telecommunications Science Foundation (Grant No. NY219114). The author wishes to express his heartfelt thanks to the referee for careful reading.
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Tan, J. Carleson Measure Spaces with Variable Exponents and Their Applications. Integr. Equ. Oper. Theory 91, 38 (2019). https://doi.org/10.1007/s00020-019-2536-0
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DOI: https://doi.org/10.1007/s00020-019-2536-0