• Abstract

In this paper,  the parabolic Schrodinger operator  $\mathcal{P}=\partial_t-\Delta +V(x)$  on $\mathbb{R}^{n+1}$ is considered, where  $n\geq 3,$  nonnegative potential $V$  belongs to the reverse H\"older class  $RH_q$  with  $q\geq n/2.$  The  $L^p$  boundedness of operators $V^{\alpha}\mathcal{P}^{-\beta},$ $ V^\alpha \nabla \mathcal{P}^{-\beta}$  and their  adjoint operators are  established.

• General Summary

• Abstract

Let $\Omega$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution to the problem

where $\partial^\pm\Omega$ represents the outer and inner boundaries of $\Omega$, respectively, $c$ is a constant. Let $\mu_k$ denote the $k$th Neumann eigenvalue of the Laplacian on $\Omega$ and $\Omega_h$ is the hole. We establish that if $\mu<\mu_8$, then $\Omega$ is an annulus.

• General Summary

• Abstract

We consider the notions of $\nu$-separation and $(N,\nu)$-separation for increasing sequences that tend to infinity. We study several of the connections between the properties of a multiplier in the space $\mathcal{S}(\mathbb{R})$ and in other related spaces and the separation properties of the sequence of its zeros.

 
We also prove that a distributional division problem

always has tempered solutions $h$ for any tempered data $f$ if and only if the non integrable function $1/F$ admits regularizations that are tempered, and that this holds if and only if the pseudofunction $\mathcal{P}f(1/F)$ is tempered.

• General Summary