We study the realization $A$ of the operator $\mathcal{A}=\frac{1}{2}\mathrm{△}-(DU,D\cdot )$ in $L^2\left(\mathrm{\Omega },\mu \right)$, where $\mathrm{\Omega }$ is a possibly unbounded convex open set in ${\mathbb{R}}^N$, $U$ is a convex unbounded function such that ${lim}_{x\to \partial \mathrm{\Omega },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$ and ${lim}_{\left|x\right|\to +\mathrm{\infty },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$, $DU\left(x\right)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu \left(dx\right)=c\exp \left(-2U\left(x\right)\right)dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D\left(A\right)=\left\{u\in H^2\left(\mathrm{\Omega },\mu \right):\left(DU,Du\right)\in L^2\left(\mathrm{\Omega },\mu \right)\right\}$ is a dissipative self-adjoint operator in $L^2\left(\mathrm{\Omega },\mu \right)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities...

Strong asymptotic completeness is shown for a pair of Schrödinger type operators on a cylindrical Lipschitz domain. A key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology. The results are based on a refined version of Mourre’s method within the context of pseudo-selfadjoint operators.

Let $A_1,\cdots ,A_m$ be $n\times n$ real matrices such that for each $1\le i\le m,$ $A_i$ is invertible and $A_i-A_j$ is invertible for $i\ne j$. In this paper we study integral operators of the form $$Tf\left(x\right)=\int k_1(x-A_{1}y)k_2(x-A_{2}y)\cdots k_m(x-A_{m}y)f\left(y\right)\mathrm{d}y,$$ $k_i\left(y\right)={\sum }_{j\in \mathbb{Z}}{2}^{jn/q_i}{\varphi }_{i,j}\left(2^{j}y\right)$, $1\le q_i<\infty ,$ $1/q_1+1/q_2+\cdots +1/q_m=1-r,$ $0\le r<1,$ and ${\varphi }_{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:H^p\left({ℝ}^n\right)\to L^q\left({ℝ}^n\right)$ for $0<p<1/r$ and $1/q=1/p-r.$ We also show that we can not expect the $H^p$-$H^q$ boundedness of this kind of operators.