This paper is a summary of the research on the mapping properties of singular integrals with rough kernels and the corresponding maximal operators. More
precisely, the author presents some recent progress, interesting problems and typical methods used in the theory concerning the boundedness and continuity for the
rough singular integral operators and maximal singular integral operators along certain submanifolds such as polynomial mappings, polynomial curves, homogeneous
mappings, surfaces of revolution and real-analytic submanifolds on the Lebesgue
spaces, Triebel–Lizorkin spaces, Besov spaces and mixed radial-angular spaces.
The main aim of this paper is to derive some new summation theorems
for terminating and truncated Clausen's hypergeometric series with unit argument,
when one numerator parameter and one denominator parameter are negative integers.
Further, using our truncated summation theorems, we obtain the Mellin transforms of
the product of exponential function and Goursat's truncated hypergeometric function.
In this paper, we derive the interior gradient estimate for solutions to general prescribed curvature equations. The proof is based on a fundamental observation
of Gårding's cone and some delicate inequalities under a suitably chosen coordinate chart. As an application, we obtain a Liouville type theorem.
In this note we show that the general theory of vector valued singular integral operators of Calderόn-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the dyadic fractional Laplacian. In doing so, we define partial derivatives in terms of Haar multipliers
and dyadic homogeneous singular integral operators.
The article considers the controllability of a diffusion equation with fractional integro-differential expressions. We prove that the resulting equation is null-controllable in arbitrary small time. Our method reduces essentially to the study of
classical moment problems.
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