Related papers: Sharp endpoint $L^p$ estimates for Schr\"odinger g…

We obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain…

We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the…

In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12],…

We consider a non-self-adjoint $h$-pseudodifferential operator $P$ in the semi-classical limit ($h\to 0$). If $p$ is the leading symbol, then under suitable assumptions about the behaviour of $p$ at infinity, we know that the resolvent…

We study the boundedness problem for maximal operators $\M$ associated to smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove that if no affine tangent plane to $S$ passes through the origin and $S$ is analytic,…

We are concerned with the two-power nonlinear Schr\"odinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^n)$ with real-valued potential $V$ for $n > 4$ and let $H_0=-\Delta$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

We give a certain $L^{\infty}(\mathbb{R}^2)$-estimate for the heat semigroup $\{e^{t\Delta}\}_{t \ge 0}$ that is closely related to the fact $H^1(\mathbb{R}^2) \not\subset L^{\infty}(\mathbb{R}^2)$, i.e., the critical Sobolev…

Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation…

We prove a sharp $L^p$-Sobolev regularity results for a class of generalized Radon transforms for families of curves in a three dimensional manifold, with folding canonical relations. The proof relies on decoupling inequalities by Wolff and…

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive…

We present a unified and concise method for establishing L^p Hardy and Rellich inequalities for a broad class of subelliptic operators of divergence type. The approach, based on a fundamental algebraic identity, provides explicit control on…

This paper is a contribution to semiclassical analysis for abstract Schr\"odinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $\mu$ be a Radon measure on $X$ with a full support.…

We study $L^p$-Sobolev improving for averaging operators $A_{\gamma}$ given by convolution with a compactly supported smooth density $\mu_{\gamma}$ on a non-degenerate curve. In particular, in 4 dimensions we show that $A_{\gamma}$ maps…

We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $\Omega\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of…

We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing point nonlinearity, $$ (\delta\text{NLS}) \qquad i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, $$ where $\delta=\delta(x)$ is the delta function…

Using sharp global heat kernel bounds and geodesic comparison geometry, we show that the Dalang condition for well-posedness of the parabolic Anderson model with measure-valued initial conditions, first introduced on Euclidean space, holds…

We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if…

This paper deals with the $L_p$-spectrum of Schr\"odinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are…

The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the…