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Concrete Operators

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Banach algebra of the Fourier multipliers on weighted Banach function spaces

Alexei Karlovich
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  • Centro de Matemática e Aplicações, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829–516 Caparica, Portugal
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Published Online: 2015-03-10 | DOI: https://doi.org/10.1515/conop-2015-0001

Abstract

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L(ℝ) is that MX,w(ℝ) is a Banach algebra.

Keywords : Fourier convolution operator; Fourier multiplier; Banach function space; Cauchy singular integral operator; rearrangement-invariant space; variable Lebesgue space; Muckenhoupt-type weight

References

  • [1] Bennett C., Sharpley R., Interpolation of Operators. Pure and Applied Mathematics, 129. Academic Press, Boston, 1988. DOI: 10.1016/S0079-8169(08)60845-4 CrossrefGoogle Scholar

  • [2] Berezhnoi E.I., Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces. Proc. Amer. Math. Soc. 127, 1999, 79-87. DOI: 10.1090/S0002-9939-99-04998-9 CrossrefGoogle Scholar

  • [3] Berkson E., Gillespie T.A., Multipliers for weighted Lp-spaces, transference, and the q-variation of functions. Bull. Sci. Math., 1998, 122, 427–454. DOI: 10.1016/S0007-4497(98)80002-X CrossrefGoogle Scholar

  • [4] Böttcher A., Karlovich Yu. I., Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics (Boston, Mass.) 154. Birkhäuser, Basel, 1997. DOI: 10.1007/978-3-0348-8922-3 CrossrefGoogle Scholar

  • [5] Böttcher A., Karlovich Yu.I., Spitkovsky I.M., Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Applications, 131. Birkhäuser, Basel, 2002. DOI: 10.1007/978-3-0348-8152-4 CrossrefGoogle Scholar

  • [6] Böttcher A., Silbermann B., Analysis of Toeplitz Operators. 2nd edn. Springer, Berlin, 2006. DOI: 10.1007/3-540-32436-4 CrossrefGoogle Scholar

  • [7] Cruz-Uribe D., Diening L., Hästö P., Themaximal operator on weighted variable Lebesgue spaces. Frac. Calc. Appl. Anal., 14, 2011, 361–374. DOI: 10.2478/s13540-011-0023-7 CrossrefGoogle Scholar

  • [8] Cruz-Uribe D., Fiorenza A., Variable Lebesgue Spaces. Birkhäuser, Basel, 2013. DOI: 10.1007/978-3-0348-0548-3 CrossrefGoogle Scholar

  • [9] Cruz-Uribe D., Fiorenza A., Neugebauer C.J., Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl., 2012, 394, 744–760. DOI: 10.1016/j.jmaa.2012.04.044 CrossrefGoogle Scholar

  • [10] Curbera G.P., García-Cuerva J., Martell J.M., Pérez C., Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 203, 2006, 256–318. DOI: 10.1016/j.aim.2005.04.009 CrossrefGoogle Scholar

  • [11] Diening L., Harjulehto P., Hästö P., Ružicka M., Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, 2017. Springer, Berlin, 2011. DOI: 10.1007/978-3-642-18363-8 CrossrefGoogle Scholar

  • [12] Duduchava R., Integral Equations with Fixed Singularities. Teubner Verlagsgesellschaft, Leipzig, 1979. Google Scholar

  • [13] Fremlin D.H., Measure Theory. Vol. 2: Broad Foundations, Torres Fremlin, Colchester, 2003. Google Scholar

  • [14] Grafakos L., Classical Fourier Analysis. 3rd ed. Graduate Texts in Mathematics, 249. Springer, New York, NY, 2014. DOI: 10.1007/978-1-4939-1194-3 CrossrefGoogle Scholar

  • [15] Grafakos L., Modern Fourier Analysis. 3rd ed. Graduate Texts in Mathematics, 250. Springer, New York, NY, 2014. DOI: 10.1007/978-1-4939-1230-8 CrossrefGoogle Scholar

  • [16] Hunt R.,Muckenhoupt B., Wheeden R., Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc., 1973, 176, 227–251. DOI: 10.1090/S0002-9947-1973-0312139-8 CrossrefGoogle Scholar

  • [17] Karlovich A.Yu., Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights. J. Oper. Theory, 2002, 47, 303–323. Google Scholar

  • [18] Karlovich A.Yu., Spitkovsky I.M., The Cauchy singular integral operator on weighted variable Lebesgue spaces. In Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, Birkhäuser, Basel. Operator Theory: Advances and Applications, 2014, 236, pp. 275–291. DOI: 10.1007/978 − 3 − 0348 − 0648 − 017 CrossrefGoogle Scholar

  • [19] Lacey M., Carleson’s theorem: proof, complements, variations. Publ. Mat., 2004, 48, 251–307. DOI: 10.5565/PUBLMAT4820401 CrossrefGoogle Scholar

  • [20] Mastylo M., Pérez C., The Hardy-Littlewood maximal type operators between Banach function spaces. Indiana Univ. Math. J., 61, 2012, 883–900. DOI: 10.1512/iumj.2012.61.4708 CrossrefGoogle Scholar

  • [21] Roch S., Santos P.A., Silbermann B., Non-Commutative Gelfand Theories. A Tool-Kit for Operator Theorists and Numerical Analysts. Universitext. Springer-Verlag London, London, 2011. DOI: 10.1007/978-0-85729-183-7 CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2015-01-28

Accepted: 2015-02-23

Published Online: 2015-03-10


Citation Information: Concrete Operators, ISSN (Online) 2299-3282, DOI: https://doi.org/10.1515/conop-2015-0001.

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© 2015 Alexei Karlovich. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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