Related papers: Dispersive estimates for principally normal pseudo…
We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes $S_q$ with $0<q\le 1$. The estimates are applied to derive semi-classical bounds for operators with smooth or…
In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and H\"ormander. We provide local stability estimates that…
In this paper, we establish a boundary observability estimate for stochastic Schr\"{o}dinger equations by means of the global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic…
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary…
In this paper, we enlarge the space of uniformly supported pseudo-differential operators on some groupoids by considering kernels satisfying certain asymptotic estimates. We show that such enlarged space contains the compact parametrix, and…
We prove dispersive estimates for the wave and Schrodinger groups associated to a second-order elliptic self-adjoint operator depending on a semi-classical parameter. Applications are made to non-trapping metric perturbations and to…
In this paper we study unique continuation theorems for magnetic Schr\"odinger equation via Carleman estimates. We use integration by parts techniques in order to show these estimates. We consider electric and magnetic potentials with…
By using some deep tools from microlocal analysis, J. Le Rousseau and L. Robbiano (Invent. Math., 183 (2011), 245--336) established several Carleman estimates for parabolic operators with isotropic diffusion coefficients which have jumps at…
We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we…
In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…
Dispersion is a fundamental concept in statistics, yet standard approaches - especially via stochastic orders - face limitations in the discrete setting. In particular, the classical dispersive order, well-established for continuous…
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 pseudospectral properties of general pseudodifferential operators around a doubly characteristic point and provide necessary and sufficient conditions for semiclassical hypoelliptic a priori estimates with a big loss of…
In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth…
In this paper we establish some new $L^{2}-L^{2}$ Carleman estimates for the Baouendi-Grushin operators $\mathscr{B}_\gamma$, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique…
We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of…
In this paper, we prove a Carleman estimate for fully-discrete approximations of parabolic operators in which the discrete parameters $h$ and $\triangle t$ are connected to the large Carleman parameter. We use this estimate to obtain…
In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result in [BL] and the arguments in [DcFLVW], we present an elementary…
We discuss properties of $L^2$-eigenfunctions of Schr\"odinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce…
The paper studies generalized differentiability properties of the marginal function of parametric optimal control problems of semilinear elliptic partial differential equations. We establish upper estimates for the regular and the limiting…