Abstract
We show that if the Hardy–Littlewood maximal operator is bounded on a separable Banach function space \(X({\mathbb {R}})\) and on its associate space \(X'({\mathbb {R}})\), then the space \(X({\mathbb {R}})\) has an unconditional wavelet basis. This result extends previous results by Soardi (Proc Am Math Soc 125:3669–3673, 1997) for rearrangement-invariant Banach function spaces with nontrivial Boyd indices and by Fernandes et al. (Banach Center Publ 119:157–171, 2019) for reflexive Banach function spaces. We specify our result to the case of Lorentz spaces \(L^{p,q}({\mathbb {R}},w)\), \(1<p<\infty \), \(1\le q<\infty \) with Muckenhoupt weights \(w\in A_p({\mathbb {R}})\).
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References
Álvarez, J., Pérez, C.: Estimates with \(A_\infty \) weights for various singular integral operators. Boll. Unione Mat. Ital. VII. Ser. A 8, 123–133 (1994)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)
Böttcher, A., Karlovich, Y.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Birkhäuser, Basel (1997)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Birkhäuser, Basel (2013)
Cwikel, M.: The dual of weak \(L^p\). Ann. Inst. Fourier 25, 81–126 (1975)
Fernandes, C.A., Karlovich, A.Y., Karlovich, Y.I.: Algebra of convolution type operators with continuous data on Banach function spaces. Banach Center Publ. 119, 157–171 (2019)
García-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam (1985)
Gut, A.: Probability: A Graduate Course. Springer, Berlin (2005)
Hernández, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)
Izuki, M., Nakai, E., Sawano, Y.: Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent. Ann. Acad. Sci. Fenn. Math. 40, 551–571 (2015)
Karlovich, A.Y., Lerner, A.K.: Commutators of singular integrals on generalized \(L^p\) spaces with variable exponent. Publ. Mat. 49, 111–125 (2005)
Karlovich, A., Shargorodsky, E.: When does the norm of a Fourier multiplier dominate its \(L^\infty \) norm? Proc. Lond. Math. Soc. 118, 901–941 (2019)
Karlovich, A.Y., Spitkovsky, I.M.: The Cauchy singular integral operator on weighted variable Lebesgue spaces. Oper. Theor. Adv. Appl. 236, 275–291 (2014)
Karlovich, Y.I., Loreto Hernández, J.: Wiener-Hopf operators with slowly oscillating matrix symbols on weighted Lebesgue spaces. Integr. Equ. Oper. Theory 64, 203–237 (2009)
Kashin, B.S., Saakyan, A.A.: Orthogonal Series, 2nd edn. Izdatel’stvo Nauchno-Issledovatel’skogo Aktuarno-Finansovogo Tsentra (AFTs), Moscow (1999). in Russian
Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral Operators in Non-standard Function Spaces. Volume 1: Variable Exponent Lebesgue and Amalgam Spaces. Birkhäuser, Basel (2016)
Lerner, A.K.: Weighted norm inequalities for the local sharp maximal function. J. Fourier Anal. Appl. 10, 465–474 (2004)
Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1995)
Meyer, Y., Coifman, R.: Wavelets: Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge (1997)
Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)
Rabinovich, V.S., Samko, S.G.: Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Integr. Equ. Oper. Theory 60, 507–537 (2008)
Soardi, P.: Wavelet bases in rearrangement invariant function spaces. Proc. Am. Math. Soc. 125, 3669–3673 (1997)
Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)
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Communicated by Sorina Barza.
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This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Projects UIDB/MAT/ 00297/2020 (Centro de Matemática e Aplicações).
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Karlovich, A.Y. Wavelet Bases in Banach Function Spaces. Bull. Malays. Math. Sci. Soc. 44, 1669–1689 (2021). https://doi.org/10.1007/s40840-020-01024-4
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DOI: https://doi.org/10.1007/s40840-020-01024-4