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.
Highlights
The solvability theory of singular integral operators has developed independently in Hölder and Lebesgue spaces [1]-[7], as norms in these spaces differ widely in their structure.The norm in weighted Hölder spaces is defined in the following way
In this work, we describe a class of operators with local singularities for which we were able to find inequalities that connect the norms in weighted Lebesgue spaces with the norms in weighted Hölder spaces
Such operators can be used in the study of boundedness, of belonging of some operators to Banach algebras and of the solvability of operators in weighted Hölder spaces, on the basis of known results for operators in weighted Lebesgue spaces
Summary
The solvability theory of singular integral operators has developed independently in Hölder and Lebesgue spaces [1]-[7], as norms in these spaces differ widely in their structure. The norm in weighted Hölder spaces is defined in the following way. (2014) Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted Hölder Spaces. In this work, we describe a class of operators with local singularities for which we were able to find inequalities that connect the norms in weighted Lebesgue spaces with the norms in weighted Hölder spaces. By way of representatives of such types of operators we may consider the following operators with local singularities: Such operators can be used in the study of boundedness, of belonging of some operators to Banach algebras and of the solvability of operators in weighted Hölder spaces, on the basis of known results for operators in weighted Lebesgue spaces
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