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We establish the monotonicity of the principal eigenvalue $\lambda_1(A)$, as a function of the advection amplitude $A$, for the elliptic operator $L_{A}=-\mathrm{div}(a(x)\nabla)+A\mathbf{V}\cdot\nabla +c(x)$ with incompressible flow…

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and…

We prove the $L^p$-boundedness of the strong maximal operator defined on a Heisenberg group w.r.t an absolutely continuous measure satisfying the product $A_\infty$-property.

In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=\Delta^2-V=\Delta^2-c|x|^{-4}$ where $c$ is any constant such that…

We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schr\"odinger operators $-\Delta + V$ with nonnegative potential $V$. These functions are proved to be bounded on $L^p$ for all $p \in (1, 2]$. The…

We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant $J$-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation…

Let $n\ge2$ and $\mathcal{L}=-\mathrm{div}(A\nabla\cdot)$ be an elliptic operator on $\mathbb{R}^n$. Given an exterior Lipschitz domain $\Omega$, let $\mathcal{L}_D$ be the elliptic operator $\mathcal{L}$ on $\Omega$ subject to the…

We consider divergence form elliptic operators of the form $L=-\dv A(x)\nabla$, defined in $R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic, complex…

We prove that wave operators of scattering theory for fourth order Schr\"odinger operators $H = \Delta^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle…

In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where $\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is…

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + f(v(x)) = 0,\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} where the matrix…

This paper is dedicated to $L^p$ bounds on eigenfunctions of a Sch\"odinger-type operator $(-\Delta_g)^{\alpha/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-\Delta_g)^{\alpha/2}$ is defined…

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove…

In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition…

In this paper, we study vector--valued elliptic operators of the form $\mathcal{L}f:=\mathrm{div}(Q\nabla f)-F\cdot\nabla f+\mathrm{div}(Cf)-Vf$ acting on vector-valued functions $f:\mathbb{R}^d\to\mathbb{R}^m$ and involving coupling at…

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We estimate simultaneously the drift and shift parameters. On the one hand, we establish a large deviation…

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +\epsilon$. The result, in particular, implies that the Stokes operator in a three-dimensional…

The aim of this paper is to study $L^p$-boundedness property of the pseudo differential operator associated with a symbol, on rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies H\"ormander-type conditions…

We show that the absolute values of non-positive eigenvalues of Schr\"odinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan, and Davies to higher…

We present some new results on analytic Ornstein-Uhlenbeck semigroups and use them to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck semigroups on open domains O to the non-symmetric case. Denoting the generator of the…