Abstract
Let \(p(\cdot ):\mathbb {R}\rightarrow (1,\infty )\) be a sufficiently regular variable exponent and \(\varrho\) be a Khvedelidze weight on \(\mathbb {R}\). Suppose that a function a belongs to the algebra \(PC_{p(\cdot ),\varrho }\) of piecewise continuous Fourier multipliers on the weighted variable Lebesgue space \(L^{p(\cdot )}(\mathbb {R},\varrho )\). We show that the Fourier convolution operator \(W^0(a)=F^{-1}aF\) is invertible on the space \(L^{p(\cdot )}(\mathbb {R},\varrho )\) if and only if its symbol a is invertible in \(L^\infty (\mathbb {R})\).
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References
C. Bennett and R. Sharpley. Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988.
E. Berkson and T. A. Gillespie. Multipliers for weighted \(L^p\)-spaces, transference, and the \(q\)-variation of functions. Bull. Sci. Math., 122(6):427–454, 1998.
A. Böttcher, Y. I. Karlovich, and I. M. Spitkovsky. Convolution operators and factorization of almost periodic matrix functions, volume 131 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2002.
N. Bourbaki. Functions of a real variable. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004. Elementary theory, Translated from the 1976 French original [MR0580296] by Philip Spain.
Y. A. Brudnyĭ and N. Y. Krugljak. Interpolation functors and interpolation spaces. Vol. I, volume 47 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa, With a preface by Jaak Peetre.
A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24:113–190, 1964.
D. Cruz-Uribe, L. Diening, and P. Hästö. The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal., 14(3):361–374, 2011.
D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl., 394(2):744–760, 2012.
D. V. Cruz-Uribe and A. Fiorenza. Variable Lebesgue spaces. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg, 2013. Foundations and harmonic analysis.
L. Diening, P. Harjulehto, P. Hästö, and M. Růžička. Lebesgue and Sobolev spaces with variable exponents, volume 2017 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011.
R. Duduchava. Integral equations with fixed singularities. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1979.
C. Fernandes, A. Karlovich, and M. Valente. Invertibility of Fourier convolution operators with piecewise continuous symbols on Banach function spaces. Trans. A. Razmadze Math. Inst., 175(1):49–61, 2021.
G. B. Folland. Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley-Interscience Publication.
I. Gohberg and N. Krupnik. One-dimensional linear singular integral equations. I, volume 53 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1992. Introduction, Translated from the 1979 German translation by Bernd Luderer and Steffen Roch and revised by the authors.
L. Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014.
A. Karlovich. Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces. In Function spaces in analysis, volume 645 of Contemp. Math., pages 165–178. Amer. Math. Soc., Providence, RI, 2015.
A. Karlovich. Algebras of continuous Fourier multipliers on variable Lebesgue spaces. Mediterr. J. Math., 17(4):Paper No. 102, 19 pp., 2020.
A. Karlovich and E. Shargorodsky. When does the norm of a Fourier multiplier dominate its \(L^\infty\) norm? Proc. Lond. Math. Soc. (3), 118(4):901–941, 2019.
A. Karlovich and E. Shargorodsky. Algebras of convolution type operators with continuous data do not always contain all rank one operators. Integral Equations Operator Theory, 93(2):Paper No. 16, 24 pp., 2021.
A. Y. Karlovich and I. M. Spitkovsky. The Cauchy singular integral operator on weighted variable Lebesgue spaces. In Concrete operators, spectral theory, operators in harmonic analysis and approximation, volume 236 of Oper. Theory Adv. Appl., pages 275–291. Birkhäuser/Springer, Basel, 2014.
B. V. Khvedelidze. Linear discontinuous boundary problems in the theory of functions, singular integral equations and some of their applications. Akad. Nauk Gruzin. SSR. Trudy Tbiliss. Mat. Inst. Razmadze, 23:3–158, 1956.
V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko. Integral operators in non-standard function spaces. Vol. 1, volume 248 of Operator Theory: Advances and Applications. Birkhäuser/Springer, [Cham], 2016. Variable exponent Lebesgue and amalgam spaces.
V. Kokilashvili, V. Paatashvili, and S. Samko. Boundedness in Lebesgue spaces with variable exponent of the Cauchy singular operator on Carleson curves. In Modern operator theory and applications, volume 170 of Oper. Theory Adv. Appl., pages 167–186. Birkhäuser, Basel, 2007.
S. G. Kreĭn, Y. I. Petunīn, and E. M. Semënov. Interpolation of linear operators, volume 54 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs.
W. A. J. Luxemburg. Banach function spaces. Thesis, Technische Hogeschool te Delft, 1955.
L. Maligranda. Orlicz spaces and interpolation, volume 5 of Seminários de Matemática [Seminars in Mathematics]. Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989.
V. Rabinovich and S. Samko. Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Integral Equations Operator Theory, 60(4):507–537, 2008.
Acknowledgements
We would like to thank the anonymous referees for useful remarks improving the presentation.
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This work was supported by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within the scope of the project UIDB/00297/2020 (Centro de Matemática e Aplicações).
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Dedicated to Professor Stefan Samko on the occasion of his 80th birthday.
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Fernandes, C., Karlovych, O. & Medalha, S. INVERTIBILITY OF FOURIER CONVOLUTION OPERATORS WITH PC SYMBOLS ON VARIABLE LEBESGUE SPACES WITH KHVEDELIDZE WEIGHTS. J Math Sci 266, 419–434 (2022). https://doi.org/10.1007/s10958-022-05897-7
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DOI: https://doi.org/10.1007/s10958-022-05897-7
Keywords
- Variable Lebesgue space
- Khvedelidze weight
- Fourier convolution operator
- Fourier multiplier
- Piecewise continuous function
- Invertibility