Trace and extension operators for fractional Sobolev spaces with variable exponent

We show that, under certain regularity assumptions, there exists a linear extension operator

On extensions of Sobolev functions defined on regular subsets of metric measure spaces

We characterize the restrictions of first order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator

Extension operators via semigroups

The Roper--Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is $A$-spirallike for a variety of linear operators $A$. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball

On Burenkov's extension operator preserving Sobolev-Morrey spaces on Lipschitz domains

We prove that Burenkov's Extension Operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space.Comment: To appear in Mathematische Nachrichte

The Fourier extension operator on large spheres and related oscillatory integrals

We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal $L^p(mathbb{S}^2)\to L^q(R \mathbb{S}^2)$ estimates for the Fourier extension operator on large spheres in $\mathbb{R}^3$, which are uniform in the radius $R$. Two appendices are included, one concerning an application to Lorentz space bounds for averaging operators along curves in $R^3$, and one on bilinear estimates