Abstract
In this paper we are deal with the generalized translation operator generated by the Bessel operator in variable exponent Lebesgue spaces. The behavior of this generalized translation operator is well known on weighted Lebesgue spaces. But, there are some differences in the behavior of these operators on the variable exponent Lebesgue spaces. For example, the generalized translation operator is bounded in the variable exponent Lebesgue space \(L_{p(\cdot ),\gamma }(\mathbb{R}^{n}_{+})\) if and only if the exponent is constant. The aim of this paper is to give some the regularity conditions which ensure the boundedness of generalized translation operator \(T^{y} \) on variable exponent Lebesgue spaces if \(p\) is nonconstant.
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References
Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Springer, Heidelberg (2013)
Diening, L.: Maximal function on generalized Lebesque spaces \(L^{p(\cdot )}\). Math. Inequal. Appl. 7, 245–253 (2004)
Diening, L., Harjulehto, P., Hästö, P., Lebesgue, R.M.: And Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)
Diening, L., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn., Math. 34, 503–522 (2009)
Ekincioglu, I.: The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted \(L_{p,\omega ,\gamma }\)-spaces with general weights. Acta Appl. Math. 109(2), 591–598 (2010)
Ekincioglu, I., Ozkin, I.K.: On high order Riesz transformations generated by generalized shift operator. Turk. J. Math. 21, 51–60 (1997)
Ekincioglu, I., Serbetci, A.: On weighted estimates of high-order Riesz-Bessel transformations generated by the generalized shift operator. Acta Math. Sin. 21(1), 53–64 (2005)
Ekincioglu, I., Kaya, E., Guliyev, S.V.: \(B_{n} \)-Maximal operator and \(B_{n} \)- singular integral operators on variable exponent Lebesgue spaces. Math. Slovaca 70(4), 893–902 (2020)
Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)
Guliyev, V.S.: On maximal function and fractional integral, associated with the Bessel differential operator. Math. Inequal. Appl. 6(2), 317–330 (2003)
Kipriyanov, I.A.: Singular Elliptic Boundary Value Problems. Nauka, Moscow (1997). (In Russian)
Klyuchantsev, M.I.: On singular integrals generated by the generalized shift operator I. Sib. Zh. Vychisl. Mat. 11, 810–821 (1970). English translation: Siberian Math. J. 11:612–620
Kovacik, O., Rökosn, J.: On spaces \(L_{p(x)}\) and \(W_{1,p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)
Levitan, B.M.: Expansion in Fourier series and integrals with Bessel functions. Usp. Mat. Nauk 6(2(42)), 102–143 (1951)
Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3, 200–211 (1931)
Ragusa, M.A., Tachikawa, A.: On minimizers for functionals under the non-standard growth conditions. AIP Conf. Proc. 1738, 480112 (2016)
Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9, 710–728 (2020)
Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)
Serbetci, A., Ekincioǧlu, I.: Boundedness of Riesz potential generated by generalized shift operator on Ba spaces. Czechoslov. Math. J. 54(3), 579–589 (2004)
Sharapudinov, I.I.: On the uniform boundedness in \(L^{p}\) (\(p=p(x)\)) of some families of convolution operators. Mat. Zametki 59(2), 291–302 (1996). 320
Shishkina, E.L., Transmutations, S.S.M.: Singular and Fractional Differential Equations with Applications to Mathematical Physics. Elsevier, Amsterdam (2020)
Tsenov, I.V.: Generalization of the problem of best approximation of a function in the space \(L^{s}\). Uch. Zap. Dagestan. Gos. Univ. 7, 25–37 (1961)
Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5(1), 105–116 (1998)
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The first author was also partially supported by the Grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455).
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Ekincioglu, I., Shishkina, E.L. & Kaya, E. On the Boundedness of the Generalized Translation Operator on Variable Exponent Lebesgue Spaces. Acta Appl Math 173, 4 (2021). https://doi.org/10.1007/s10440-021-00411-8
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DOI: https://doi.org/10.1007/s10440-021-00411-8