Related papers: Carleman estimates for higher step Grushin operato…

Let $\Delta_{\Omega_\varepsilon}$ be the Dirichlet Laplacian in the domain $\Omega_\varepsilon:=\Omega\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $\Omega\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny…

We study the Calder\'on problem for a logarithmic Schr\"odinger type operator of the form $L_{\Delta} +q$, where $L_{\Delta}$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-\Delta)^s$ of the…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on…

The Agmon estimate shows that eigenfunctions of Schr\"odinger operators, $ -\Delta \phi + V \phi = E \phi$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $\left\{x: V(x) > E…

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…

In this paper, we introduce a new sequence of operators based on the Gr\"unwald interpolation operators on Chebyshev nodes on the space $L^p[0,{\pi}]$. The operators we consider are integral variants of the Gr\"unwald interpolation…

The Sklyanin algebra $S_{\eta}$ has a well-known family of infinite-dimensional representations $D(\mu)$, $\mu \in C^*$, in terms of difference operators with shift $\eta$ acting on even meromorphic functions. We show that for generic…

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[ \mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} +\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}% \] where $(a_{ij})…

We establish the kernel estimates for the Littlewood-Paley projections associated with a Schr\"odinger operator H=-\Delta+V in \mathbb{R}^3 for a large class of short-range potentials V(x). As a corollary, we prove the homogeneous Sobolev…

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schr\"odinger operator $\Delta +V$ with potential $V\in C^{1,\alpha}(M_0)$ for some $\alpha>0$, the Dirichlet-to-Neumann map $N|_{\Gamma}$ measured on…

We prove a gradient estimate for Donaldson's equation \[\omega\wedge(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n-1}=e^F(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^n\] (and its parabolic analog) on an $n$-dimensional compact…

Let $L= -\Delta+ V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse H\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a…

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…

We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov-Hausdorff stability. We use these results to construct pairs of Riemannian…

We give Feffermain-Stein type inequalities related to mixed estimates for Calder\'on-Zygmund operators. More precisely, given $\delta>0$, $q>1$, $\varphi(z)=z(1+\log^+z)^\delta$, a nonnegative and locally integrable function $u$ and $v\in…

We investigate the Dirichlet problem associated to the Schr\"odinger operator $\mathcal L=-\Delta_{\mathbb{H}^n}+V$ on Heisenberg group $\mathbb H^n$: \begin{align*} \begin{cases} \partial_{ss}u(g,s)-\mathcal L u(g,s)=0\,,\quad &{\rm in \,\…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ be the operator norm…

In this note we study the property of unique continuation for solutions of $|(-\Delta)^{\alpha/2}u|\leq|Vu|$, where $V$ is in a function class of potentials including $\bigcup_{p>n/\alpha}L^p(\mathbb{R}^n)$ for $n-1\leq\alpha<n$. In…

We consider the Schur-Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is…

We consider pseudodifferential operators on functions on $\R^{n+1}$ which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a…