Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted Hölder Spaces

Affiliation(s)

Institute of Basic Sciences and Engineering, Hidalgo State University, Pachuca, Mexico.

Institute of Basic Sciences and Engineering, Hidalgo State University, Pachuca, Mexico.

ABSTRACT

In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted Hölder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted Hölder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted Hölder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted Hölder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

KEYWORDS

Endpoint Singularities, Weighted Holder Space, Weighted Lebesgue Spaces, Relation between Norms, Boundedness

Endpoint Singularities, Weighted Holder Space, Weighted Lebesgue Spaces, Relation between Norms, Boundedness

Cite this paper

Karelin, O. and Tarasenko, A. (2014) Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted Hölder Spaces.*Applied Mathematics*, **5**, 2779-2785. doi: 10.4236/am.2014.517266.

Karelin, O. and Tarasenko, A. (2014) Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted Hölder Spaces.

References

[1] Gakhov, F.D. (1977) Boundary Value Problems. Nauka, Moscow. (in Russian)

[2] Muskhelishvili, N.I. (2008) Singular Integral Equations, Boundary Value Problems of the Theory of Functions and Some of Their Applications to Mathematical Physics. Dover Publications, Mineola.

[3] Litvinchuk, G.S. (2000) Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic Publishers, Dordrecht, Boston, London.

http://dx.doi.org/10.1007/978-94-011-4363-9

[4] Gohberg, I. and Krupnik, N. (1992) One-Dimensional Linear Singular Integral Equations. Operator Theory: Advances and Applications Vol. 53. Birkhauser Verlag, Basel, Boston, Berlin.

[5] Mikhlin, S.G. and Prossdorf, S. (1986) Singular Integral Operators. Akademie-Verlag, Berlin.

http://dx.doi.org/10.1007/978-3-642-61631-0

[6] Duduchava, R.V. (1963) Unidimensional Singular Integral Operator Algebras in Spaces of Holder Functions with Weight. Proceedings of A. Razmadze Mathematical Institute, 43, 19-52. (in Russian)

[7] Karapetiants, N.K. and Samko, S.G. (2001) Equations with Involutive Operator. Birkhauser Verlag, Boston, Basel, Berlin.

http://dx.doi.org/10.1007/978-1-4612-0183-0

[8] Duduchava, R.V. (1979) Convolution Integral Equations with Discontinuous Presymbols, Singular Integral Equations with Fixed Singularities and Their Applications to Problem in Mechanics. Proceedings of A. Razmadze Mathematical Institute, 60, 2-136. (in Russian)

[9] Karlovich, Yu. and Kravchenko, V. (1981) Singular Integral Equations with Non-Carleman Shift on an Open Contour. Differential Equations, 17, 1408-1417.

[10] Kravchenko, V.G. and Litvinchuk, G.S. (1994) Introduction to the Theory of Singular Integral Operators with Shift. Kluwer Academic Publishers, Dordrecht, Boston, London.

http://dx.doi.org/10.1007/978-94-011-1180-5

[11] Karelin, A. (1980) On a Boundary Value Problem with Shift for a System of Differential Equations of Elliptichyperbolic Type. Soviet Mathematics-Doklady, 22, 507-512.

[1] Gakhov, F.D. (1977) Boundary Value Problems. Nauka, Moscow. (in Russian)

[2] Muskhelishvili, N.I. (2008) Singular Integral Equations, Boundary Value Problems of the Theory of Functions and Some of Their Applications to Mathematical Physics. Dover Publications, Mineola.

[3] Litvinchuk, G.S. (2000) Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic Publishers, Dordrecht, Boston, London.

http://dx.doi.org/10.1007/978-94-011-4363-9

[4] Gohberg, I. and Krupnik, N. (1992) One-Dimensional Linear Singular Integral Equations. Operator Theory: Advances and Applications Vol. 53. Birkhauser Verlag, Basel, Boston, Berlin.

[5] Mikhlin, S.G. and Prossdorf, S. (1986) Singular Integral Operators. Akademie-Verlag, Berlin.

http://dx.doi.org/10.1007/978-3-642-61631-0

[6] Duduchava, R.V. (1963) Unidimensional Singular Integral Operator Algebras in Spaces of Holder Functions with Weight. Proceedings of A. Razmadze Mathematical Institute, 43, 19-52. (in Russian)

[7] Karapetiants, N.K. and Samko, S.G. (2001) Equations with Involutive Operator. Birkhauser Verlag, Boston, Basel, Berlin.

http://dx.doi.org/10.1007/978-1-4612-0183-0

[8] Duduchava, R.V. (1979) Convolution Integral Equations with Discontinuous Presymbols, Singular Integral Equations with Fixed Singularities and Their Applications to Problem in Mechanics. Proceedings of A. Razmadze Mathematical Institute, 60, 2-136. (in Russian)

[9] Karlovich, Yu. and Kravchenko, V. (1981) Singular Integral Equations with Non-Carleman Shift on an Open Contour. Differential Equations, 17, 1408-1417.

[10] Kravchenko, V.G. and Litvinchuk, G.S. (1994) Introduction to the Theory of Singular Integral Operators with Shift. Kluwer Academic Publishers, Dordrecht, Boston, London.

http://dx.doi.org/10.1007/978-94-011-1180-5

[11] Karelin, A. (1980) On a Boundary Value Problem with Shift for a System of Differential Equations of Elliptichyperbolic Type. Soviet Mathematics-Doklady, 22, 507-512.