Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo

Sharp constants in weighted trace inequalities on Riemannian manifolds

We establish some sharp weighted trace inequalities W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M) on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of $M$ and $\sigma\in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.Comment: 34 page

Dual complements for domains of C n

Let Ω ⊂ C n be a bounded, strictly convex domain and Ω ∼ be its dual complement. Very few such domains with fully described dual complements have been known. We present new types of domains for which their dual complements can be completely described. © 2019 Element D.O.O.. All rights reserved

Remarks on compactness results for variable exponent spaces Lp(⋅)

Given the Lebesgue space with variable exponent Ls(⋅)(Ω) whose norm is denoted by ||⋅||s(⋅), we show the following equivalence: lim|E|→0⁡||χE||s(⋅)=0 if and only if [Formula presented], where χE is the characteristic function of the measurable set E and |E| its Lebesgue measure. We apply such results to characterize compactness of some inclusions