Related papers: Convex Hypersurfaces and $L^p$ Estimates for Schr\…

In this paper we consider the space-fractional Schr\"odinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper…

The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite dimensional complex analysis, but this is a…

In this paper, we prove $L^2 \to L^p$ estimates, where $p>2$, for spectral projectors on a wide class of hyperbolic surfaces. More precisely, we consider projections in small spectral windows $[\lambda-\eta,\lambda+\eta]$ on geometrically…

We study integro-differential elliptic equations (of order $2s$) with variable coefficients, and prove the natural and most general Schauder-type estimates that can hold in this setting, both in divergence and non-divergence form.…

In this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption. We obtain the optimal $H^s$-$L^q$ estimate and its mixed norm generalization. As applications we prove some weighted…

Let $L=-\Delta + V(x)$ be a Schr\"odinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz…

We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the…

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For $L$ in some class of elliptic operators, we study weighted norm $L^p$ inequalities for singular…

We consider parabolic Schr\"odinger type equations associated to fractional powers of uniformly elliptic 2m-order operators with constant coefficients. Potentials and initial data are considered in suitable Morrey spaces. By means of…

In this article we study for $p\in (1,\infty)$ the $L^p$-realization of the vector-valued Schr\"odinger operator $\mathcal{L}u := \mathrm{div} (Q\nabla u) + V u$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and…

Consider operators $L_{V}:=\Delta + V$ in a bounded smooth domain $D$ in $R^N$. Assume that $V\in C^1(D)$ and $V$ may blow up at the boundary at most as $1/\delta^2$ where $\delta$ denotes distance to the boundary. Assume also that $L_{V}$…

We prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ Sobolev improvement theorems for local averaging operators over real analytic surfaces in ${\mathbb R}^3$. For most such operators, in a sense made precise in the paper, the set of…

In the whole space $R^d$ ($d\ge 2$), we study homogenization of a divergence-form matrix elliptic operator $L_\varepsilon$ of an arbitrary even order larger than 2 with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is…

Given a complex, elliptic coefficient function we investigate for which values of $p$ the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly…

In this article we study the Schr\"odinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm estimates…

In this paper we prove that, under suitable assumptions on {\alpha} > 0, the operator L = (1 + |x|{\alpha})\Delta admits realizations generating contraction or analytic semigroups in Lp (RN). For some values of {\alpha}, we also explicitly…

Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schr\"odinger term: $-\Delta_p u + \mathbb{V}|u|^{p-2}u$ with bound constraints $\psi_1…

Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we consider the associated wave operators W_+, W_- defined as the strong L^2 limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the wave…

In this paper we study the rate of convergence of the eigenvalues of 1-dimensional rapidly oscillating $p-$laplacian type problems and find explicit order of convergence both in $k$ and in $\ve$. Moreover, explicit bounds on the constant…

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is…