Searcharxiv⌕ Search

arXiv subjects

El Maati Ouhabaz

Publications and source records attributed to El Maati Ouhabaz.

At least 19 recordsLinked to original sources
arXiv2026

${\mathcal R}$-boundedness of Riesz transforms and spectral multipliers of Schrödinger operators

Let $\LL = -Δ+ V$ be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform $\nabla \LL^{-\sfrac{1}{2}}$ is bounded on $L^p(\RR^d)$ for $p \in (1,2]$, with a bound independent of $V$. We prove more, namely that the set \[ \{ \nabla \LL^{-\sfrac{1}{2}},\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) \} \] is ${\mathcal R}$-bounded on $L^p(\RR^d)$ for $p \in (1,2]$. We also prove the ${\mathcal R}$-boundedness on $L^p(\RR^d)$, for $p \in (1,\infty)$, of the spectral multipliers \[ { m(\LL),\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) } \] for any $m:[0, \infty) \to \CC$ satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators $-{\bf Δ} + {\bf V}$ on vector-valued spaces $L^p(\RR^d,\ell^2)$. In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.

math.AP↗
arXiv2026

Weak type $(1,1)$ estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\LL = ({\bf d}+ iA)^*({\bf d}+iA) + V$ where $A$ is a real $1$-differential form in $L^2_{loc}$ and $V \in L^1_{loc}$ is a nonnegative electric potential. We prove that the magnetic Riesz transform $({\bf d} + iA) \LL^{-\einhalb}$ is of weak type $(1,1)$ with bound at most $2$. It extends to a bounded operator on $L^p$ for $p \in (1, 2]$ with norm bounded by a constant $c_p$ depending only on $p$. It is then independent of the dimension of the manifold and independent of $A$ and $V$. We also prove weak type $(1,1)$ bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of $\RR^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.

math.AP↗
arXiv2026

Maximal inequalities and Riesz transforms for vector-valued magnetic Schrödinger operators

We consider vector-valued magnetic Schrödinger operators $-\bm Δ_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries belong to $L^1_{\mathrm{loc}}(\mathbb{R}^d)$. We prove maximal inequalities in $L^p(\mathbb{R}^d;\mathbb{C}^m)$, $p\in[1,\infty)$ and the boundedness of the Riesz transforms $(\nabla - i\bm a)(-\bm Δ_{\bm a}+V)^{-\frac{1}{2}}$ and $V^α(-\bm Δ_{\bm a}+V)^{-α}$ on $L^p(\mathbb{R}^d;\mathbb{C}^m)$ for every $p \in (1,2]$ and every $α\in[0,1/p]$.

math.AP↗
arXiv2025

Lower bounds for the spectral gap and an extension of the Bonnet-Myers theorem

On a fairly general class of Riemannian manifolds M, we prove lower estimates in terms of the Ricci curvature for the spectral bound (when M has infinite volume) and for the spectral gap (when M has finite volume) for the Laplace-Beltrami operator. As a byproduct of our results we obtain an extension of the Bonnet-Myers theorem on the compactness of the manifold. We also prove lower bounds for the spectral gap for Ornstein-Uhlenbeck type operators on weighted manifolds. As an application we prove lower bounds for the spectral gap of perturbations of some radial measures on R n .

math.AP↗
arXiv2023

Fractional anisotropic Calderón problem on complete Riemannian manifolds

We prove that the metric tensor $g$ of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator. This later is associated to a fractional power of the Laplace-Belrami operator $Δ_g$. Our result holds under the condition that the metric tensor $g$ is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in \cite{FGKU}

math.AP↗
arXiv2023

A simple construction of the Anderson operator via its quadratic form in dimensions two and three

We provide a simple construction of the Anderson operator in dimensions two and three. This is done through its quadratic form. We rely on an exponential transform instead of the regularity structures or paracontrolled calculus which are usually used for the construction of the operator. The knowledge of the form is robust enough to deduce important properties such as positivity and irreducibility of the corresponding semigroup. The latter property gives existence of a spectral gap.

math.AP↗
arXiv2023

A Lyapunov approach to stability of positive semigroups: An overview with illustrations

The stability analysis of possibly time varying positive semigroups on non necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, physics and molecular chemistry. This article presents an overview of some recent Lyapunov-based approaches, focusing principally on practical and powerful tools for designing Lyapunov functions. These techniques include semigroup comparisons as well as conjugacy principles on non necessarily bounded manifolds with locally Lipschitz boundaries. All the Lyapunov methodologies discussed in the article are illustrated in a variety of situations, ranging from conventional Markov semigroups on general state spaces to more sophisticated conditional stochastic processes possibly restricted to some non necessarily bounded domains, including locally Lipschitz and smooth hypersurface boundaries, Langevin diffusions as well as coupled harmonic oscillators.

math.PR↗
arXiv2022

Littlewood-Paley-Stein functionals: an R-boundedness approach

Let $L = Δ+ V$ be a Schrödinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the vertical Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^p(M)$ {\it if and only if} the set $\{\sqrt{t}\, \nabla e^{-tL}, \, t > 0\}$ is ${\mathcal R}$-bounded on $L^p(M)$. We also introduce and study more general functionals. For a sequence of functions $m_k : [0, \infty) \to \mathbb{C}$, we define $$H((f_k)) := \Big( \sum_k \int_0^\infty | \nabla m_k(tL) f_k |^2 dt \Big)^{1/2} + \Big( \sum_k \int_0^\infty | \sqrt{V} m_k(tL) f_k |^2 dt \Big)^{1/2}.$$ Under fairly reasonable assumptions on $M$ we prove boundedness of $H$ on $L^p(M)$ in the sense $$\| H((f_k)) \|_p \le C\, \Big\| \Big( \sum_k |f_k|^2 \Big)^{1/2} \Big\|_p$$ for some constant $C$ independent of $(f_k)_k$. A lower estimate is also proved on the dual space $L^{p'}$. We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper.

math.AP↗
arXiv2022

Interpolation for analytic families of multilinear operators on metric measure spaces

Let (X j , d j , $μ$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $κ$ $\le$ $\infty$ for $κ$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $\bullet$ $\bullet$ $\bullet$ L p m (X m) $\rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Stein's complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators T z are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schr{ö}dinger operators on L p is included.

math.AP↗
arXiv2021

Bernstein inequalities via the heat semigroup

We extend the classical Bernstein inequality to a general setting including Schr{ö}dinger operators and divergence form elliptic operators on Riemannian manifolds or domains. Moreover , we prove a new reverse inequality that can be seen as the dual of the Bernstein inequality. The heat kernel will be the backbone of our approach but we also develop new techniques such as semi-classical Bernstein inequalities, weak factorization of smooth functions {à} la Dixmier-Malliavin and BM O -- L $\infty$ multiplier results (in contrast to the usual L $\infty$ -- BM O ones). Also, our approach reveals a link between the L p-Bernstein inequality and the boundedness on L p of the Riesz transform. The later being an important subject in harmonic analysis. 2010 Mathematics Subject Classifications: 35P20, 58J50, 42B37 and 47F05.

math.AP↗
arXiv2021

The square root of a parabolic operator

Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $τ$) be a uniformly elliptic operator with boundary conditions on a domain $Ω$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on L 2 (0, $τ$, L 2 ($Ω$)) by (Lu)(t) := $\partial$u(t) $\partial$t + L(t)u(t). We assume a very little of regularity for the boundary of $Ω$ and assume that the coefficients A(x, t) are measurable in x and piecewise C $α$ in t for some $α$ > 1 2. We prove the Kato square root property for $\sqrt$ L and the estimate $\sqrt$ L u L 2 (0,$τ$,L 2 ($Ω$)) $\approx$ $\nabla$ x u L 2 (0,$τ$,L 2 ($Ω$)) + u H 1 2 (0,$τ$,L 2 ($Ω$)) + $τ$ 0 u(t) 2 L 2 ($Ω$) dt t 1/2. We also prove L p-versions of this result. Keywords: elliptic and parabolic operators, the Kato square root property, maximal regularity, the holomorphic functional calculus, non-autonomous evolution equations.

math.AP↗
arXiv2020

The diamagnetic inequality for the Dirichlet-to-Neumann operator

Let $Ω$ be a bounded domain in R d with Lipschitz boundary $Γ$. We define the Dirichlet-to-Neumann operator N on L 2 ($Γ$) associated with a second order elliptic operator A = -- d k,j=1 $\partial$ k (c kl $\partial$ l) + d k=1 b k $\partial$ k -- $\partial$ k (c k $\times$) + a 0. We prove a criterion for invariance of a closed convex set under the action of the semigroup of N. Roughly speaking, it says that if the semigroup generated by --A, endowed with Neumann boundary conditions, leaves invariant a closed convex set of L 2 ($Ω$), then the 'trace' of this convex set is invariant for the semigroup of N. We use this invariance to prove a criterion for the domination of semigroups of two Dirichlet-to-Neumann operators. We apply this criterion to prove the diamagnetic inequality for such operators on L 2 ($Γ$).

math.AP↗
arXiv2019

The Kato square root problem on an arbitrary domain of $\mathbb{R}^d$

We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow perturbations by general complex potentials.

math.AP↗
arXiv2019

Sharp spectral multipliers without semigroup framework and application to random walks

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates à la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results. Finally, we consider the random walk on the integer lattice $\mathbf{Z}^n$ and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on $\mathbb{R}^n$.

math.AP↗
arXiv2017

Lions' maximal regularity problem with H 1/ 2 -regularity in time

We consider the problem of maximal regularity for non-autonomous Cauchy problems u ' (t) + A(t) u(t) = f (t), t $\in$ (0, $τ$ ] u(0) = u 0. The time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We are interested in J.L. Lions's problem concerning maximal regularity of such equations. We give a positive answer to this problem under minimal regularity assumptions on the forms. Our main assumption is that the forms are piecewise H 1 2 with respect to the variable t. This regularity assumption is optimal and our results are the most general ones on this problem.

math.AP↗
arXiv2017

Dirichlet-to-Neumann and elliptic operators on C 1+$κ$ -domains: Poisson and Gaussian bounds

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable H{ö}lder coefficients when the underlying domain is bounded and has a C 1+$κ$-boundary for some $κ$ > 0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish H{ö}lder continuity of $\nabla$ x $\nabla$ y G up to the boundary. These results are used to prove L p-estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C 1+$κ$-domains. Such estimates are the keystone in our approach for the Poisson bounds.

math.AP↗
arXiv2017

L p -estimates for the heat semigroup on differential forms, and related problems

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let -- $\rightarrow$ $Δ$ k be the Hodge-de Rham Laplacian on differential k-forms with k $\ge$ 1. By the Bochner decomposition formula -- $\rightarrow$ $Δ$ k = * + R k. Under the assumption that the negative part R -- k is in an enlarged Kato class, we prove that for all p $\in$ [1, $\infty$], e --t -- $\rightarrow$ $Δ$ k p--p $\le$ C(t log t) D 4 (1-- 2 p) (for large t). This estimate can be improved if R -- k is strongly sub-critical. In general, (e --t -- $\rightarrow$ $Δ$ k) t>0 is not uniformly bounded on L p for any p = 2. We also prove the gradient estimate e --t$Δ$ p--p $\le$ Ct -- 1 p , where $Δ$ is the Laplace-Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on L p for p > 2.

math.AP↗