Related papers: Unique continuation for the heat operator with pot…

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global H\"ormander classes of order $-n(1-\rho)/2$ on $G\times \widehat{G}$. Our approach…

In this paper, the authors define the weak Herz spaces and the weak Herz-type Hardy spaces with variable exponent. As applications, the authors establish the boundedness for a large class of singular integral operators including some…

Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\R^d)$, when…

In this paper, we establish a fundamental inequality for fourth order partial differential operator $\cal P=\alpha\partial_s+\beta\partial_{ss}+\Delta^2$ ($\alpha, \beta\in\mathbb{R}$) with an abstract exponential-type weight function. Such…

We study the existence and nonexistence of singular solutions to the equation $u_t-\Delta u - \frac{\kappa}{|x|^2}u+|x|^\alpha u|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is,…

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected…

In this paper, we study Hardy-type uncertainty principles and unique continuation properties for linear covariant Schrodinger equations with variable coefficients in the presence of bounded electric and magnetic potentials. Under suitable…

We prove an inequality of H\"older type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness…

This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschtiz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by…

We investigate the quantitative unique continuation properties of solutions to second-order elliptic equations with lower-order terms. In particular, we establish quantitative forms of the strong unique continuation property for solutions…

For the Schr\"odinger operator $-\Delta_\rm{g}+V$ on a complete Riemannian manifold with real valued potential $V$ of compact support, we establish a sharp equivalence between Sobolev regularity of $V$ and the existence of finite-order…

We consider Hardy operators, i.e., homogeneous Schr\"odinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential, which only depends on the appropriate power of the distance to the boundary of the…

We investigate uniqueness in the inverse problem of reconstructing simultaneously a spacewise conductivity function and a heat source in the parabolic heat equation from the usual conditions of the direct problem and additional information…

We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $\Omega \subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods…

In this work, we study the unique continuation properties of Robin boundary value problems with Robin potentials $\eta \in L_{d-1+\varepsilon}$. Our results generalize earlier ones in which $\eta$ was assumed to be either zero (Neumann…

In this paper we explore the weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces. For this we use the Fourier series expansions. Moreover,…

We obtain Sobolev inequalities for the Schrodinger operator -\Delta-V, where V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply these inequalities to obtain pointwise estimates on the associated heat kernel,…

We prove the strong unique continuation property for many-body Schr\"odinger operators with an external potential and an interaction potential both in $L^p_{\rm loc}(\mathbb{R}^d)$, where $p > \max(2d/3,2)$, independently of the number of…

We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t -\operatorname{div}(A(x) \nabla_x)^s + q(x,t)$ for $s\in(0,1)$ and show the unique recovery of $q$ from exterior measured data.…

In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded…