On the boundedness of the generalized translation operator on variable exponent Lebesgue spaces. (English) Zbl 1466.47024
Summary: 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.
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 45P05 | Integral operators |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 47G10 | Integral operators |
Keywords:
generalized translation operator; Laplace-Bessel operator; variable exponent Lebesgue spacesReferences:
| [1] | Chen, Y.; Levine, S.; Rao, R., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 1383-1406 (2006) · Zbl 1102.49010 |
| [2] | Cruz-Uribe, D.; Fiorenza, A., Variable Lebesgue Spaces (2013), Heidelberg: Springer, Heidelberg · Zbl 1268.46002 |
| [3] | Diening, L., Maximal function on generalized Lebesque spaces \(L^{p(\cdot )}\), Math. Inequal. Appl., 7, 245-253 (2004) · Zbl 1071.42014 |
| [4] | Diening, L.; Harjulehto, P.; Hästö, P.; Lebesgue, R. M., And Sobolev Spaces with Variable Exponents (2011), Berlin: Springer, Berlin · Zbl 1222.46002 |
| [5] | 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) · Zbl 1180.42010 |
| [6] | 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) · Zbl 1192.42012 |
| [7] | Ekincioglu, I.; Ozkin, I. K., On high order Riesz transformations generated by generalized shift operator, Turk. J. Math., 21, 51-60 (1997) · Zbl 0882.44002 |
| [8] | 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) · Zbl 1093.47049 |
| [9] | 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) · Zbl 1512.42023 |
| [10] | Frazier, M.; Jawerth, B., Decomposition of Besov spaces, Indiana Univ. Math. J., 34, 4, 777-799 (1985) · Zbl 0551.46018 |
| [11] | Guliyev, V. S., On maximal function and fractional integral, associated with the Bessel differential operator, Math. Inequal. Appl., 6, 2, 317-330 (2003) · Zbl 1047.47035 |
| [12] | Kipriyanov, I. A., Singular Elliptic Boundary Value Problems (1997), Moscow: Nauka, Moscow · Zbl 0892.35002 |
| [13] | Klyuchantsev, M. I., On singular integrals generated by the generalized shift operator I, Sib. Zh. Vychisl. Mat., 11, 810-821 (1970) · Zbl 0204.12601 |
| [14] | Kovacik, O.; Rökosn, J., On spaces \(L_{p(x)}\) and \(W_{1,p(x)}\), Czechoslov. Math. J., 41, 592-618 (1991) · Zbl 0784.46029 |
| [15] | Levitan, B. M., Expansion in Fourier series and integrals with Bessel functions, Usp. Mat. Nauk, 6, 2-42, 102-143 (1951) · Zbl 0043.07002 |
| [16] | Orlicz, W., Über konjugierte Exponentenfolgen, Stud. Math., 3, 200-211 (1931) · JFM 57.0251.02 |
| [17] | Ragusa, M. A.; Tachikawa, A., On minimizers for functionals under the non-standard growth conditions, AIP Conf. Proc., 1738 (2016) |
| [18] | Ragusa, M. A.; Tachikawa, A., Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9, 710-728 (2020) · Zbl 1420.35145 |
| [19] | Ružička, M., Electrorheological Fluids: Modeling and Mathematical Theory (2000), Berlin: Springer, Berlin · Zbl 0968.76531 |
| [20] | Serbetci, A.; Ekincioǧlu, I., Boundedness of Riesz potential generated by generalized shift operator on Ba spaces, Czechoslov. Math. J., 54, 3, 579-589 (2004) · Zbl 1080.47503 |
| [21] | 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) · Zbl 0873.47023 |
| [22] | Shishkina, E. L.; Transmutations, S. S.M., Singular and Fractional Differential Equations with Applications to Mathematical Physics (2020), Amsterdam: Elsevier, Amsterdam · Zbl 1454.35003 |
| [23] | 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) |
| [24] | Zhikov, V. V., On some variational problems, Russ. J. Math. Phys., 5, 1, 105-116 (1998) · Zbl 0917.49006 |
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