New aspects in the weight theory and applications

Abstract: In our lecture we are going to present a survey of recent results and new aspects in weight theory. The discussion will concern the following problems:

\item{1.} Solutions of two-weight problems for integral transforms with positive kernel and maximal functions on homogeneous type spaces.

\item{2.} Criteria for weighted Orlicz class inequalities for maximal functions and singular integrals on homogeneous type spaces.

\item{3.} Two-weight estimates for singular integrals on homogeneous type groups and multipliers of Fourier transforms.

\item{4.} Applications to the Dirichlet and Riemann-Hilbert problems for domains with non-smooth boundaries.

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