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On the regularization of a multidimensional integral equation in Lebesgue spaces with variable exponent. (English) Zbl 1278.45003
Math. Notes 93, No. 4, 583-592 (2013); translation from Mat. Zametki 93, No. 4, 575-585 (2013).
The first-kind multidimensional integral equation with potential-type kernel of small order is considered \[ \mathbf{M}^{\alpha}\varphi:=\int_{\mathbb{R}^n} \frac{c(x,y)}{|x-y|^{n-\alpha}}\varphi(y)dy=f(x), \quad x\in\mathbb{R}^n, \tag{1} \] where \(0<\alpha<1\) and the function \(c(x,y)\) satisfies the following conditions:
\[ c(x,y)\in C(\mathbb{R}^n\times \mathbb{R}^n),\quad c(x,x)\in L^{\infty}(\mathbb{R}^n), \quad \inf\limits_{x\in\mathbb{R}^n}|c(x,x)|>0; \]

\[ |c(x,y)-c(z,y)|\leq \frac{C|x-z|^{\lambda}}{(1+|x|)^{\lambda}(1+|z|)^{\lambda}}, \quad \alpha<\lambda\leq 1, \] where \(C>0\) is independent of \(x,y,z\).
The integral equation (1) can be reduced to an equation of the second kind by regularization of the form \(\mathbf{RM}^{\alpha}\varphi=\varphi+\mathbf{A}\varphi\), where \(\mathbf{R}\) is some hypersingular operator, \(\mathbf{A}\) is an operator compact in \(L^p(\mathbb{R}^n)\), \(1<p<n/\alpha\).
The authors extend this result to the case of Lebesque spaces \(L^{p(\cdot)}(\mathbb{R}^n)\) with variable exponent.
MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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[1] С. М. Умархаджиев, Многомерные интегральные уравнения первогорода с ядром типа потенциала, Деп. в ВИНИТИ \?1743-81, Ростовский гос. ун-т, 1981
[2] С. М. Умархаджиев, Исследование многомерных операторов типа потенциала с непрерывными и разрывными характеристиками, Дис. \cdots канд. физ.-матем. наук, Ростовский гос. ун-т, 1982
[3] S. G. Samko, Hypersingular Integrals and their Applications, Anal. Methods Spec. Funct., 5, Taylor & Francis, London, 2002 · Zbl 0998.42010
[4] И. И. Шарапудинов, “О топологии пространства \(\mathscr L^{p(t)}([0,1])\)”, Матем. заметки, 26:4 (1979), 613 – 632 · Zbl 0437.46024 · doi:10.1007/BF01159546 · mi.mathnet.ru
[5] O. Kovać\?ik, J. Raḱosniḱ, “On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\)”, Czechoslovak Math. J., 41(116):4 (1991), 592 – 618 · Zbl 0784.46029 · eudml:13956
[6] L. Diening, M. Ruz\?ic\?ka, “Calderoń – Zygmund operators on generalized Lebesgue spaces \({L}^{p( \cdot )}\) and problems related to fluid dynamics”, J. Reine Angew. Math., 563 (2003), 197 – 220 · Zbl 1072.76071 · doi:10.1515/crll.2003.081
[7] M. Ruz\?ic\?ka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., 1748, Springer-Verlag, Berlin, 2000 · Zbl 0962.76001 · doi:10.1007/BFb0104029
[8] В. В. Жиков, “Усреднение функционалов вариационного исчисления и теории упругости”, Изв. АН СССР. Сер. матем., 50:4 (1986), 675 – 710 · Zbl 0599.49031 · doi:10.1070/IM1987v029n01ABEH000958 · mi.mathnet.ru
[9] L. Diening, P. Ha\"sto\", A. Nekvinda, “Open problems in variable exponent Lebesgue and Sobolev spaces”, Function Spaces, Differential Operators and Nonlinear Analysis (Bohemian-Moravian Uplands, May 28 – June 2, 2004), Math. Inst. Acad. Sci. Czech Republick, Prague, 2005, 38 – 58
[10] V. Kokilashvili, “On a progress in the theory of integral operators in weighted Banach function spaces”, Function Spaces, Differential Operators and Nonlinear Analysis (Bohemian-Moravian Uplands, May 28 – June 2, 2004), Math. Inst. Acad. Sci. Czech Republick, Prague, 2005, 152 – 175
[11] V. Kokilashvili, S. Samko, “Weighted boundedness of the maximal, singular and potential operators in variable exponent spaces”, Analytic Methods of Analysis and Differential Equations, Cambridge Sci. Publ., Cambridge, 2008, 139 – 164 · Zbl 1151.42006
[12] S. Samko, “On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators”, Integral Transforms Spec. Funct., 16:5-6 (2005), 461 – 482 · Zbl 1069.47056 · doi:10.1080/10652460412331320322
[13] X. Fan, D. Zhao, “On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\)”, J. Math. Anal. Appl., 263:2 (2001), 424 – 446 · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617
[14] S. G. Samko, “Differentiation and integration of variable order and the spaces \({L}^{p(x)}\)”, Operator Theory and Complex and Hypercomplex Analysis (Mexico City, December 12 – 17, 1994), Contemp. Math., 212, Amer. Math. Soc., Providence, RI, 1998, 203 – 219 · Zbl 0958.26005
[15] М. А. Красносельский, “О теореме М. Рисса”, ДАН СССР, 131 (1960), 246 – 248 · Zbl 0097.10202
[16] F. Cobos, T. Kuḧn, T. Schonbek, “One-sided compactness results for Aronsjain – Gagliardo functors”, J. Funct. Anal., 106:2 (1992), 274 – 313 · Zbl 0787.46061 · doi:10.1016/0022-1236(92)90049-O
[17] M. Cwikel, “Real and complex interpolation and extrapolation of compact operators”, Duke Math. J., 65:2 (1992), 333 – 343 · Zbl 0787.46062 · doi:10.1215/S0012-7094-92-06514-8
[18] K. Hayakawa, “Interpolation by the real method preserves compactness of operators”, J. Math. Soc. Japan, 21 (1969), 189 – 199 · Zbl 0181.13703 · doi:10.2969/jmsj/02120189
[19] J.-L. Lions, J. Peetre, “Sur une classe d/espaces d/interpolation”, Inst. Hautes Eťudes Sci. Publ. Math., 19 (1964), 5 – 66 · Zbl 0148.11403 · doi:10.1007/BF02684796 · numdam:PMIHES_1964__19__5_0 · eudml:103841
[20] A. Persson, “Compact linear mappings between interpolation spaces”, Ark. Mat., 5 (1964), 215 – 219 · Zbl 0128.35204 · doi:10.1007/BF02591123
[21] V. Rabinovich, S. Samko, “Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces”, Integral Equations Operator Theory, 60:4 (2008), 507 – 537 · Zbl 1153.47043 · doi:10.1007/s00020-008-1566-9
[22] S. Samko, “On compactness of operators in variable exponent Lebesgue spaces”, Topics in Operator Theory, Vol. 1, Oper. Theory Adv. Appl., 202, Birkhaüser Verlag, Basel, 2010, 497 – 507 · Zbl 1204.47044
[23] S. G. Samko, “Hardy inequality in the generalized Lebesgue spaces”, Fract. Calc. Appl. Anal., 6:4 (2003), 355 – 362 · Zbl 1098.46022
[24] V. Kokilashvili, S. Samko, “The maximal operator in weighted variable exponent spaces on metric spaces”, Georgian Math. J., 15:4 (2008), 683 – 712 · Zbl 1167.42312 · www.heldermann.de
[25] S. Samko, E. Shargorodsky, B. Vakulov, “Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. II”, J. Math. Anal. Appl., 325:1 (2007), 745 – 751 · Zbl 1107.47016 · doi:10.1016/j.jmaa.2006.01.069
[26] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, C. Perez, “The boundedness of classical operators on variable \(L^p\) spaces”, Ann. Acad. Sci. Fenn. Math., 31:1 (2006), 239 – 264 · Zbl 1100.42012 · eudml:126704
[27] А. Almeida, “Inversion of the Riesz potential operator on Lebesgue spaces with variable exponent”, Frac. Calc. Appl. Anal., 6:3 (2003), 311 – 327 · Zbl 1094.42009
[28] А. Almeida, S. Samko, “Characterization of Riesz and Bessel potentials on variable Lebesgue spaces”, J. Funct. Spaces Appl., 4:2 (2006), 113 – 144 · Zbl 1129.46022 · doi:10.1155/2006/610535
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