Related papers: The $L^p$-dissipativity of certain differential an…
We give complete algebraic characterizations of the $L^{p}$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$, were ${\mathscr A}^{hk}(x)$…
Various notions of dissipativity type for partial differential operators and their applications are surveyed. We deal with functional dissipativity and its particular case $L^p$-dissipativity. Most of the results are due to the authors.
We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}\xi,\xi>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}\xi,\xi>$ (for any $\xi\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the…
The basic purpose of the present paper is the full solutions of the inverse problem (i.e. a finding of necessary and sufficient conditions) for the operator with complex periodic coefficients.
The optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of singular integral operators of both Calder\'on-Zygmund and of rough type are shown in the different exponent ranges $``q>p"$, $``q=p"$ and $``q<p"$ to…
In the present paper we consider the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$ (${\mathop{\mathscr…
In this paper, we prove $L^p$ decay estimates for multilinear oscillatory integrals in $\mathbb{R}^2$, establishing sharpness through a scaling argument. The result in this paper is a generalization of the previous work by Gressman and Xiao…
We determine necessary and sufficient conditions on the ring of differential operators of a finite purely inseparable field extension of positive characteristic for determining whether the extension is modular.
In this paper, we consider $L^p$- estimate for a class of oscillatory integral operators satisfying the Carleson-Sj\"olin conditions with further convex and straight assumptions. As applications, the multiplier problem related to a general…
We show that every operator on $L^{p}$, $1<p<\infty$ defined by multiplication by the identity function on $\mathbb{C}$ is a compact perturbation of an operator that is diagonal with respect to an unconditional basis. We also classify these…
In this note we investigate the two notions of expansivity and strong structural stability for composition operators on $L^p$ spaces, $1 \leq p < \infty$. Necessary and sufficient conditions for such operators to be expansive are provided,…
The purpose of this paper is to obtain an integral representation for the difference $f(L_1)-f(L_2)$ of functions of maximal dissipative operators. This representation in terms of double operator integrals will allow us to establish…
We discuss the closability of directional derivative operators with respect to a general Radon measure $\mu$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from…
We establish an almost sharp L^r to L^p estimate for oscillatory integral operators satisfying the cinematic curvature condition. The proof combines Wolff's two-ends reduction with refined decoupling inequalities.
In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are…
We prove an $L^p$-version of the limiting absoprtion principle for a class of periodic elliptic differential operators of second order. The result is applied to the construction of nontrivial solutions of nonlinear Helmholtz equations with…
We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators…
The aim of the article is to prove $L^{p}-L^{q}$ off-diagonal estimates and $L^{p}-L^{q}$ boundedness for operators in the functional calculus of certain perturbed first order differential operators of Dirac type for with $p\le q$ in a…
The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of…
Let $A$ be a homogeneous elliptic differential operator of order $m$ on $% \Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main result is as follows: Suppose that $u\in L_{loc}^{1}$ and that $Au\in L^{p}$ for some…