arXiv subjects

M. I. Neiman-zade

Publications and source records attributed to M. I. Neiman-zade.

3 recordsLinked to original sources

arXiv2003

Multipliers from Sobolev space $H^\al_p$ into $H^{-\al}_p$

A function $q(x)$ is said to be a multiplier from the Sobolev space $H^\al_p(R^n)$ into $H^{-\al}_p(R^n)$ if the operator $Lf(x)=q(x)f(x)$ is a bounded operator from the first space into the second one. Let $M^\al_p$ the the space of such multipliers. In this paper we give the description of the spaces $M^\al_p$ provided the condition $\al>min(n/p,n/p')$. In the case $\al\le min(n/p,n/p')$ we obtain some embedding theorems for Sobolev spaces with negative smoothness indices into $M^\al_p$.

math.FA↗

arXiv2003

Strongly elliptic operators with distributional coefficients

We study operators of the form $$ L=\sum_{|α|,|β|\le n} D^αc_{α,β} D^β, x\in\mathbb R^n $$ provided that the coefficients of the main symbol corresponding to the indices $|α|=|β|=m$ are continuous while the other ones are distributions. Assuming that the main symbol defines the strongly elliptic operator we find sufficient conditions for the coefficients $c_{α,β}(x)$ which guarantee that the operator $L$ is well defined. In particular, if $c_{α,β}(x)$ belong to the spaces of multipliers from $H_2^{m-|α|}$ to $H_2^{|β|-m}$ then $L$ defines a maximal sectorial operator in $L_2(\mathbb R^n)$. We also describe the spaces of multipliers.

math.FA↗

arXiv2003

Schroedinger and elliptic operators with distributional coefficients on a bounded domain

We study the operator $L=-Δ+q$ on a bounded domain $Ω\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and generalized Neumann boundary conditions. The asymptotics of the eigenvalues and the basis properties of the eigen- and associated functions of such operators are studied. The results are generalized for strongly elliptic operator of order $2m$ with Dirichlet boundary conditions.

math.FA↗