Boundedness of Weighted Singular Integral Operators on a Carleson Curve in Grand Lebesgue Spaces

Abstract

Symmetry of extremals of functional inequalities via spectral estimates for linear operators J. Math. Phys. 53, 095204 (2012) On the structure of positive maps: Finite-dimensional case J. Math. Phys. 53, 023515 (2012) Eigenfunctions decay for magnetic pseudodifferential operators J. Math. Phys. 52, 093709 (2011) An estimate for the resolvent of a non-self-adjoint differential operator on the half-line J. Math. Phys. 52, 043515 (2011) Additional information on AIP Conf. Proc.Abstract. We obtain the necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Because of applications to singular integral equations, the underlying set where the functions are defined is a Carleson curve in the complex plane. Note that weighted boundedness of an operator in grand Lebesgue space is known to be not the same as the boundedness in weighted grand Lebesgue space.

Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces

“…The boundedness of the maximal operator and Sobolev's inequality for grand Morrey spaces with doubling measure were also studied in [32]. See also [15] and [31], etc. Our first aim in this paper is to establish the boundedness of the maximal operator on grand Morrey spaces of variable exponents over non-doubling measure spaces.…”