Related papers: Maximal Inequalities and Riesz Transform Estimates…
The main focus of this paper is commutators and maximal commutators on Orlicz spaces for fractional maximal functions and Riesz potential. The main advance in comparison with the existing results is that we manage to obtain conditions for…
We study $L^p$ bounds for two kinds of Riesz transforms on $\mathbb{R}^d$ related to the harmonic oscillator. We pursue an explicit estimate of their $L^p$ norms that is independent of the dimension $d$ and linear in $\max(p, p/(p-1))$.
The first author established in [8] a quantitative Borg-Levinson theorem for the Schr\"odinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schr\"odinger operator. We discuss both the…
For $1<p<\infty$, we establish the $L_{p}$ boundedness of the maximal Riesz transforms in terms of the Riesz transforms on quantum tori $L_{p}(\mathbb{T}^{d}_{\theta})$, and quantum Euclidean space $L_{p}(\mathbb{R}^{d}_{\theta})$. In…
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…
We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition,…
We prove global, scale invariant Strichartz estimates for the linear magnetic Schr\"odinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global…
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…
In this paper, we obtain the desired noncommutative maximal inequalities of the truncated Calder\'on-Zygmund operators of non-convolution type acting on operator-valued $L_p$-functions for all $1<p<\infty$, answering a question left open in…
In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain…
We prove optimal spectral inequalities for Landau operators in full space and in arbitrary dimension. Spectral inequalities are lower bounds on the L 2 -mass of functions in spectral subspaces of finite energy when integrated over a…
We prove optimal high-frequency resolvent estimates for perturbations by large magnetic and electric potentials
This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S^{\delta}_{\lambda}(\mathcal{L}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schr\"odinger operator $\mathcal{L}_{\mathbf{A}}$ on…
The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schr\"odinger operator $-\Delta-V(x)$ in terms of an $L^p(\mathbb{R}^d)$ norm of the potential $V$. We prove here the existence…
In this paper we prove identifiability and stability estimates for a local-data inverse boundary value problem for a magnetic Schr\"odinger operator in dimension $n\geq 3$. We assume that the inaccessible part of the boundary is part of a…
This paper present an overview of some of the applications of the martingale inequalities of D.L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms,…
We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calder\'on-Zygmund operators suitably defined on…
We study spectral properties of Schr\"odinger operators on $\RR^d$. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in $\ZZ^d$, with the property that frequencies of finite patterns are…
This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schr{\"o}dinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role…
In this paper we generalize the H\'ajek-R\'enyi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of…