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Extrapolation to weighted Morrey spaces with variable exponents and applications

Part of: Integral, integro-differential, and pseudodifferential operators Miscellaneous topics - Partial differential equations Linear function spaces and their duals Nontrigonometric harmonic analysis Harmonic analysis in several variables

Published online by Cambridge University Press:  09 November 2021

Kwok-Pun Ho Open the ORCID record for Kwok-Pun Ho [Opens in a new window]
Kwok-Pun Ho*
Affiliation:
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China(vkpho@eduhk.hk)
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Abstract

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Keywords

Morrey spacesTriebel–Lizorkin spacesHardy–Morrey spacesextrapolationpseudo-differential operatorsFourier integral operatorsCalderón–Zygmund operatorswavelets

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47G30: Pseudodifferential operators 35S30: Fourier integral operators
Secondary: 42B30: $H^p$-spaces 42C40: Wavelets and other special systems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 1002 - 1027
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Almeida, A., Hasanov, J. and Samko, S., Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J. 15 (2008), 1–15.CrossRefGoogle Scholar
Bennett, C. and Sharpley, R., Interpolations of operators (Academic Press, Orlando, 1988).Google Scholar
Bui, H.-Q., Weighted Hardy spaces, Math. Nachr. 103 (1981), 45–62.Google Scholar
Bui, H.-Q., Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), 581–605.Google Scholar
Bui, H.-Q., Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures, J. Funct. Anal. 55 (1984), 39–62.CrossRefGoogle Scholar
Bui, H.-Q., Palusyński, M. and Taibleson, M., A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), 219–246.Google Scholar
Caetanoa, A. and Kempka, H., Variable exponent Triebel-Lizorkin-Morrey spaces, J. Math. Anal. Appl. 484 (2020), 123712.CrossRefGoogle Scholar
Cruz-Uribe, D., Fiorenza, A. and Neugebauer, C., Weighted norm inequalities for the maximal operator on variable Lebesgue spaces, J. Math. Anal. Appl. 394 (2012), 744–760.CrossRefGoogle Scholar
Cruz-Uribe, D. and Fiorenza, A., Variable Lebesgue spaces: foundations and harmonic analysis (Birkhäuser/Springer, Basel, 2013).CrossRefGoogle Scholar
Deng, Y. and Long, S., Pseudodifferential operators on weighted Hardy spaces, J. Funct. Space 2020 (2020), 7154125.Google Scholar
Diening, L., Harjulehto, P., Hästö, P. and Ružička, M, Lebesgue and Sobolev spaces with variable exponent, Lecture Notes in Mathematics, Volume 2017 (Springer-Verlag, Berlin, 2011).CrossRefGoogle Scholar
Ferreira, D. and Staubach, W., Global and local regularity of Fourier integral operators on weighted and unweighted spaces, Memoirs Amer. Math. Soc. 1074 (2014), 1–65.Google Scholar
Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34–170.CrossRefGoogle Scholar
García-Cuerva, J., Weighted $H^{p}$ spaces, Dissertations Math. 162 (1979), 1–63.Google Scholar
García-Cuerva, J. and Kazarian, K., Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces, Studia Math. 109 (1994), 255–276.CrossRefGoogle Scholar
García-Cuerva, J. and Martell, J., Wavelet characterization of weighted spaces, J. Gemo. Anal. 11 (2001), 241–264.CrossRefGoogle Scholar
Grafakos, L., Modern Fourier analysis (Springer-Verlag, New York, 2009).CrossRefGoogle Scholar
Guliyev, V. and Samko, S., Maximal potential and singular operators in the generalized variable exponent Morrey spaces on unbounded sets, J. Math. Sci. (N. Y.) 193 (2013), 228–248.CrossRefGoogle Scholar
Guliyev, V., Hasanov, J. and Samko, S., Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand. 107 (2010), 285–304.CrossRefGoogle Scholar
Guliyev, V., Hasanov, J. and Badalov, X., Maximal and singular integral operators and their commutators on generalized weighted Morrey spaces with variable exponent, Math. Ineq. Appl. 21 (2018), 41–61.Google Scholar
Haroske, D. and Skrzypczak, L., Embeddings of weighted Morrey spaces, Math. Nachr. 290 (2016), 1066–1086.CrossRefGoogle Scholar
Ho, K.-P., Littlewood-Paley spaces, Math. Scand. 108 (2011), 77–102.CrossRefGoogle Scholar
Ho, K.-P., Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces, Ann. Acad. Sci. Fenn. Math. 37 (2012), 375–406.CrossRefGoogle Scholar
Ho, K.-P., Atomic decompositions of weighted Hardy-Morrey spaces, Hokkaido Math. J. 42 (2013), 131–157.CrossRefGoogle Scholar
Ho, K.-P., Atomic decomposition of Hardy-Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), 31–62.CrossRefGoogle Scholar
Ho, K.-P., Atomic decompositions of weighted Hardy spaces with variable exponents, Tohoku Math. J. (2) 69 (2017), 383–413.CrossRefGoogle Scholar
Ho, K.-P., Sublinear operators on weighted Hardy spaces with variable exponents, Forum Math. 31 (2019), 607–617.CrossRefGoogle Scholar
Ho, K.-P., Boundedness of operators and inequalities on Morrey-Banach spaces, Publ. Res. Inst. Math. Sci. (2021) (to appear).Google Scholar
Ho, K.-P., Calderón-Zygmund operators. Bochner-Riesz means and parametric Marcinkiewicz integrals on Hardy-Morrey spaces with variable exponents, Kyoto J. Math. (2021)(to appear).Google Scholar
Ho, K.-P., Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces, J. Math. Anal. Appl. 503 (2021), 125279.CrossRefGoogle Scholar
Ho, K.-P., Singular integral operators and sublinear operators on Hardy local Morrey spaces with variable exponents, Bull. Sci. Math. 171 (2021), 103033.CrossRefGoogle Scholar
Jia, H. and Wang, H., Decomposition of Hardy-Morrey spaces, J. Math. Anal. Appl. 354 (2009), 99–110.CrossRefGoogle Scholar
Jiao, Y., Zhao, T. and Zhou, d., Variable martingale Hardy-Morrey spaces, J. Math. Anal. Appl. 484 (2020), 123722.CrossRefGoogle Scholar
Kokilashvili, V. and Meskhi, A., Boundedness of maximal and singular operators in Morrey spaces with variable exponent, Armen. J. Math. 1 (2008), 18–28.Google Scholar
Kokilashvili, V. and Samko, S., Maximal and fractional operators in weighted $L^{p}(x)$ spaces, Rev. Mat. Iberoamericana 20 (2004), 493–515.CrossRefGoogle Scholar
Komori, Y. and Shirai, S., Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2009), 219–231.CrossRefGoogle Scholar
Liang, Y., Sawano, Y., Ullrich, T., Yang, D. and Yuan, W., A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces, Dissertationes Math. (Rozprawy Mat.) 489 (2013), 114.Google Scholar
Mastyło, M., Sawano, Y. and Tanaka, H., Morrey type space and its Köthe dual space, Bull. Malays. Math. Soc. 41 (2018), 1181–1198.CrossRefGoogle Scholar
Mazzucato, A., Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355 (2003), 1297–1364.CrossRefGoogle Scholar
Michalowski, N., Rule, D. and Staubach, W., Weighted norm inequalities for pseudo-differential operators defined by amplitudes, J. Funct. Anal. 258 (2010), 4183–4209.CrossRefGoogle Scholar
Miller, N., Weighted Sobolev spaces and pseudodifferential operators with smooth symbols, Trans. Am. Math. Soc. 269 (1982), 91–109.CrossRefGoogle Scholar
Mizuta, Y. and Shimomura, T., Weighted Morrey spaces of variable exponent and Riesz potentials, Math. Nachr. 288 (2015), 984–1002.CrossRefGoogle Scholar
Morrey, C. B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.CrossRefGoogle Scholar
Nakai, E., Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–103.CrossRefGoogle Scholar
Nakai, E., Sadasue, G. and Sawano, Y., Martingale Morrey-Hardy and Campanato-Hardy Spaces, J. Funct. Spaces 2013 (2013), 690258. 14 pages.Google Scholar
Nakamura, S., Generalized weighted Morrey spaces and classical operators, Math. Nachr. 289 (2016), 2235–2262.CrossRefGoogle Scholar
Nakamura, S., Noi, T. and Sawano, Y., Generalized Morrey spaces and trace operator, Science China Math. 59 (2016), 281–336.CrossRefGoogle Scholar
Nakamura, S. and Sawano, Y., The singular integral operator and its commutator on weighted Morrey spaces, Collect. Math. 68 (2017), 145–174.CrossRefGoogle Scholar
Nishigiki, S., Weighted norm inequalities for certain pseudo-differential operators, Tokyo J. Math. 7 (1984), 129–140.Google Scholar
Rubio de Francia, J., Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 393–395.CrossRefGoogle Scholar
Rubio de Francia, J., A new technique in the theory of $A_p$ weights, Topics in modern harmonic analysis, Volume I, II (Turin/Milan, 1982), pp. 571–579. Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983.Google Scholar
Rubio de Francia, J., Factorization theory and $A_p$ weights, Amer. J. Math. 106 (1984), 533–547.CrossRefGoogle Scholar
Sato, S., A note on weighted estimates for certain classes of pseudo-differential operators, Rocky Mountain J. Math. 35 (2005), 267–284.CrossRefGoogle Scholar
Sato, S., Non-regular pseudo-differential operators on the weighted Triebel-Lizorkin spaces, Tohoku Math. J. (2) 59 (2007), 323–339.CrossRefGoogle Scholar
Sawano, Y., Wavelet characterization of Besov-Morrey and Triebel-Lizorkin-Morrey spaces, Funct. Approx. Comment. Math. 38 (2008), 93–107.CrossRefGoogle Scholar
Sawano, Y., A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math. Sin. (Engl. Ser.) 25 (2009), 1223–1242.CrossRefGoogle Scholar
Sawano, Y. and Tanaka, H., Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Math. Z. 257 (2007), 871–905.CrossRefGoogle Scholar
Sawano, Y. and Tanaka, H., Predual spaces of Morrey spaces with nondoubling measures, Tokyo J. Math. 32 (2009), 471–486.CrossRefGoogle Scholar
Sawano, Y., Ho, K.-P., Yang, D. and Yang, S., Hardy spaces for ball quasi-Banach function spaces, Dissertationes Mathematicae 525 (2017), 1–102.CrossRefGoogle Scholar
Stein, E., Harmonic analysis (Real-variable methods, orthogonality, and oscillatory integrals (Princeton, NJ, Princeton University Press, 1993).Google Scholar
Strömberg, J.-O. and Torchinsky, A., Weighted Hardy spaces, Lecture Notes in Mathematics, Vol. 1381 (1989).CrossRefGoogle Scholar
Tang, L. and Xu, J., Some properties of Morrey type Besov-Triebel spaces, Math. Nachr. 278 (2005), 904–917.CrossRefGoogle Scholar
Tao, J., Yang, D. C. and Yang, D. Y., Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces, Math. Methods Appl. Sci., 42 (2019), 1631–1651.CrossRefGoogle Scholar
Torres, R., Boundedness results for operators with singular kernels on distribution spaces, Memoirs Amer. Math. Soc. 442 (1991).Google Scholar
Triebel, H., Interpolation theory, function spaces, differential operators, North-Holland Math. Library, Volume 18 (North-Holland, Amsterdam, 1978).Google Scholar
Triebel, H., Theory of function spaces, Monographs in Math., Volume 78 (Birkhäuser, Basel 1983).CrossRefGoogle Scholar
Xu, J., Variable Besov and Triebel–Lizorkin spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), 511–522.Google Scholar
Yabuta, K., Weighted norm inequalities for pseudodifferential operators, Osaka J. Math. 23 (1986), 703–723.Google Scholar
Zorko, C., Morrey spaces, Proc. Amer. Math. Soc. 98 (1986), 586–592.CrossRefGoogle Scholar