Some Remarks on Integral Operators in Banach Function Spaces and Representation Theorems in Banach-Sobolev Spaces

Year 2024, Volume: 14 Issue: 2, 189 - 204, 31.07.2024

Eminaga M. Mamedov Natavan P. Nasibova Yonca Sezer

Abstract

In this paper, we consider convolution operators, integral operators with
weak singularity, Riesz potentials, in particular, those with kernels Ki (x, y) = xi−yi|x−y|n
acting in special classes of Banach function spaces X (Ω) and their subspaces Xs (Ω)), and
we prove some representation theorems for the functions from Banach-Sobolev spaces.
We also prove the boundedness of Riesz potential in additive-invariant spaces.

Keywords

Banach function space, rearrangement-invariant space, additive-invariant, Sobolev space, integral operator, weak singularity, Riesz potential

References

• [1] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, 1988, 469 p.
• [2] D.W. Boyd, Spaces between a pair of reflexive Lebesgue spaces, Proc. Amer. Math. Soc., 18(2), 1967, 215-219.
• [3] B.T. Bilalov, S.R. Sadigova, On the Fredholmness of the Dirichlet problem for a second-order elliptic equation in grand-Sobolev spaces, Ricerche mat., 73, 2024, 283–322.
• [4] E.M. Stein, Singular operator and differentiability properties of functions, Moscow, Mir, 1973 (translation into Russian)
• [5] S.G. Mikhlin, Linear partial differential equations, Moscow, Visshaya skola, 1977 (in Russian).
• [6] B.T. Bilalov, S.R. Sadigova, On local solvability of higher order elliptic equations in rearrangement invariant spaces, Siberian Mathematical Journal, 63(3), 2022, 516-530.
• [7] E.M. Mamedov, On substitution and extension operators in Banach-Sobolev function spaces, Proceedings of the Institute of Math. and Mech., Nat. Ac. of Sciences of Azer., 48(1), 2022, 88-103.
• [8] R.E. Castillo, H. Rafeiro, An introductory course in Lebesgue spaces, Springer, 2016.
• [9] E.M. Mamedov, N.A. Ismailov, On structural theorems in Banach function spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics, 43(4), 2023, 114-127.
• [10] V.A. Solonnikov, N.N. Uraltseva, Sobolev spaces, in: Selected Topics of Higher Algebra and Analysis, Leningrad, Leningrad Gos. Univ., 1981, 129- 199 (in Russian).
• [11] B.T. Bilalov, E.M. Mamedov, Y. Sezer, N.P. Nasibova, Compactness in Banach function spaces. Poincare and Friedrichs inequalities. (submitted)
• [12] B.T. Bilalov, S.R. Sadigova, On solvability in the small of higher order elliptic equations in grand-Sobolev spaces, Complex Variables and Elliptic Equations, 66(12), 2021, 2117-2130.
• [13] B.T. Bilalov, S.R. Sadigova, Interior Schauder-type estimates for higherorder elliptic operators in grand-Sobolev spaces, Sahand Communications in Mathematical Analysis, 1(2), 2021, 129-148.
• [14] E.M. Mamedov, S. Cetin, Interior Schauder-type estimates for m-th order elliptic operators in rearrangement-invariant Sobolev spaces, Turk J Math., 48(4), 2024, 793-816.
• [15] B.T. Bilalov, T.M. Ahmadov, Y. Zeren, S.R. Sadigova, Solution in the Small and Interior Shauder-type Estimate for the m-th Order Elliptic Operator in Morrey-Sobolev Spaces, Azerb. J. Math, 12(2), 2022, 190-219.
• [16] B.T. Bilalov, S.R. Sadigova, S. Cetin, The concept of a trace and boundedness of the trace operator in Banach-Sobolev function spaces, Numerical Functional Analysis and Optimization, 43(9), 2022, 1069-1094.
• [17] B.T. Bilalov, Hardy’s Banach functional classes and methods of boundary value problems in questions of bases, Baku, Elm, 2022, 272 p. (in Russian).
• [18] S.S. Byun, D.K. Palagachev, L.G. Softova, Survey on gradient estimates for nonlinear elliptic equations in various function spaces, St. Petersburg Math. J., 31(3), 2020, 401-419 and Algebra Anal., 31(3), 2019, 10-35.
• [19] D.K. Palagachev, L.G. Softova, Elliptic Systems in Generalized Morrey Spaces, Azerb. J. Math., 11(2), 2021, 153-162.
• [20] L. Caso, R. D’Ambrosio, L. Softova, Generalized Morrey Spaces over Unbounded Domains, Azerb. J. Math., 10(1), 2020, 193-208.
• [21] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, Volume 1: Variable Exponent Lebesgue and Amalgam Spaces, Springer, 2016.
• [22] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, Volume 2: Variable Exponent H¨older, Morrey–Campanato and Grand Spaces, Springer, 2016.
• [23] D. Cruz-Uribe, O.M. Guzman, H. Rafeiro, Weighted Riesz Bounded Variation Spaces and the Nemytskii operator, Azerb J. Math., 10(2), 2020, 125-139.
• [24] R.E. Castillo, H. Rafeiro, E.M. Rojas, Unique Continuation of the Quasilinear Elliptic Equation on Lebesgue Spaces Lp, Azerb J. Math., 11(1), 2021, 136-153.