We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we obtain the sharp $L^p$ maximal bounds for the Schrödinger operators on asymptotically conic manifolds, the harmonic oscillator and its perturbations or elliptic operators on compact manifolds.
We investigate spectral multipliers, Bochner-Riesz means and convergence of eigenfunction expansion corresponding to the Schrödinger operator with anharmonic potential ${\mathcal L}=-\frac{d^2}{dx^2}+|x|$. We show that the Bochner-Riesz profile of the operator ${\mathcal L}$ completely coincides with such profile of the harmonic oscillator ${\mathcal H}=-\frac{d^2}{dx^2}+x^2$. It is especially surprising because the Bochner-Riesz profile for the one-dimensional standard Laplace operator is known to be essentially different and the case of operators ${\mathcal H}$ and ${\mathcal L}$ resembles m…
This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators $P(D)-z$ and all $z\in {\mathbb C}\backslash [0, \infty)$, which give an extension of the second order results of Kenig-Ruiz-Sogge \cite{KRS}. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schrödinger operators $P(D)+V$ with small integrable potentia…
In this paper, we study the $L^p$-boundedness of the commutator $[b, S_R^δ(H)](f) = bS_R^δ(H) f - S_R^δ(H)(bf)$ of a BMO function $b$ and the Bochner-Riesz means $S_R^δ(H)$ for Hermite operator $H=-Δ+|x|^2$ on $\mathbb{R}^n$, $n\geq2$. We show that if $δ>δ(q)=n(1/q -1/2)- 1/2$, the commutator $[b,S_R^δ(H)]$ is bounded on $L^p(\mathbb{R}^n)$ whenever $q<p<q'$ and $1\leq q\leq 2n/(n+2)$.
We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner-Riesz summability results with sharp exponent.
We obtain weak-type $(p, p)$ endpoint bounds for Bochner-Riesz means for the Hermite operator $H=-Δ+|x|^2$ in ${\mathbb R}^n, n\geq 2$ and for other related operators for $1\leq p\leq 2n/(n+2)$, extending earlier results of Thangavelu and of Karadzhov.
For $f\in {\frak S}({\Bbb R}^d)$, we consider the Bochner-Riesz operator ${\frak R}^δ$ of index $δ>0$ defined by $$\hat {{\frak R}^δf}(ξ)=(1-|ξ|^2)^δ_+ \hat f (ξ).$$ Then we prove the Bochner-Riesz conjecture which states that if $δ>\max\{d|1/p-1/2|-1/2,0\}$ and $p>1$ then ${\frak R}^δ$ is a bounded operator from $L^p({\Bbb R}^d)$ into $L^p({\Bbb R}^d)$; moreover, if $δ(p)=d(1/p-1/2)-1/2$ and $1<p<2d/(d+1)$, then ${\frak R}^{δ(p)}$ is a bounded operator from $L^p({\Bbb R}^d)$ into $L^{p,\infty}({\Bbb R}^d)$.
We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Boc…
We prove the existence of $L^2$ solutions to the Helmholtz equation $(-Δ- 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-Δ-1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schrödinger operators. The same techniques are then used to prove that a wide range of $L^p \mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their action on the clo…
We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to $(p,2)$ restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersiv…
The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to use these estimates to prove new estimates for Bochner Riesz means in this setting. Our eigenfunction estimates involve estimating the $L^2\to L^\infty$ mapping properties of the operators $χ_λ$ which project onto unit bands of the spectrum of the square root of the Laplacian. These generalize the recent estimates of D. Grieser for individual eigenfunctions.…
Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence gen…
We introduce some new functions spaces to investigate some problems at or beyond endpoint. First, we prove that Bochner-Riesz means $B_R^λ$ are bounded from some subspaces of $L^p_{|x|^α}$ to $L^p_{|x|^α}$ for $ \frac{n-1}{2(n+1)}<λ\leq \frac{n-1}{2}, 0 < p\leq p'_λ=\frac{2n}{n+1+2λ}, n(\frac{p}{p_λ}-1)< α<n(\frac{p}{p'_λ}-1)$, and $0<R<\infty,$ and so are the maximal Bochner-Riesz means $B_*^λ$ for $ \frac{n-1}{2}\leq λ< \infty, 0 < p\leq 1$ and $-n< α<n(p-1)$. From these we obtain the $L^p_{|x|^α}$-norm convergent property of $B_R^λ$ for these $λ,p,$ and $α$. Second, let $n\geq 3,$ we prove …
We establish a Riesz potential criterion for Lebesgue continuity points of functions of bounded $\mathbb{A}$-variation, where $\mathbb{A}$ is a $\mathbb{C}$-elliptic differential operator of arbitrary order. This result might even be of interest for classical functions of bounded variation.
We describe weighted Plancherel estimates and sharp Hebisch-Müller-Stein type spectral multiplier result for a new class of Grushin type operators. We also discuss the optimal exponent for Bochner-Riesz summability in this setting.
In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*} \mathrm{div}(A(x,\nabla u)) = \mathrm{div}(f) \ \text{in} \ Ω, \mbox{ and } \ u = g \ \text{on} \ \partial Ω, \end{align*} that related to Riesz potentials. Here, in our setting, $Ω\subset \mathbb{R}^n$, $n \ge 2$ is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded me…
We consider Muckenhoupt weights $w$, and define weighted Hardy spaces $H^p_{\mathcal{T}}(w)$, where $\mathcal{T}$ denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated by a second order divergence form elliptic operator $L$. In the range $0<p< 1$, we give a molecular characterization of these spaces. Additionally, in the range $p\in \mathcal{W}_w(p_-(L),p_+(L))$ we see that these spaces are isomorphic to the $L^p(w)$ spaces. We also consider the Riesz transform $\nabla L^{-\frac{1}{2}}$, associated with $L$, and show that…
For any fixed $p>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transforms associated with second order elliptic operators with real, symmetric, bounded measurable coefficients.
We prove optimal integrability results for solutions of the $p(\cdot)$-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials maps $L^1$ to variable exponent weak Lebesgue spaces.
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 restrictio…
This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.
We study the equation --div(A(x, u)) = g(x, u, u) + $μ$ where $μ$ is a measure and either g(x, u, u) $\sim$ |u| q 1 u||u| q 2 or g(x, u, u) $\sim$ |u| s 1 u + ||u| s 2. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.
In this paper, we first generalize the work of Bourgain and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same $L^p$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the $L^{\infty} \to L^q$ boundedness a…
Let $ \{A^i,E^i\} $ be the elliptic complex on a $ n $-dimensional smooth closed Riemannian manifold $X$ with the first order differential operators $ A^i $ and smooth vector bundles $ E^i $ over $X$. We consider nonlinear operator equations, associated with the parabolic differential operators $\partial_t + Δ^{i} $, generated by the Laplacians $Δ^{i} $ of the complex $ \{A^i,E^i\}$, in special Bochner-Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frechét derivative $ \mathcal{A}_i' $ of the induced nonlinear mapping is continuously inve…
We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schrödinger equations.
This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $L^p$ estimates. It appears that the case $p<2$ already treated earlier is radically different from the case $p>2$ which is new. We thus recover in a unified and coherent way many $L^p$ estimates and give further applications. The key tools from harmonic a…
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…
On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we …
Consider the second order divergence form elliptic operator $L$ with complex bounded coefficients. In general, the operators related to it (such as Riesz transform or square function) lie beyond the scope of the Calderón-Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some $L^p$ spaces. In this work we generalize the classical approach and develop a theory of Hardy and BMO spaces associated to $L$, which includes, in particular, molecular decomposition, maximal function characterization, duality of Hardy and BMO spaces, John-Nirenberg inequality, and allows to han…
We consider the abstract non-negative self-adjoint operator $L$ acting on $L^2(X)$ which satisfies Davies-Gaffney estimates and the corresponding Hardy spaces $H^p_L(X)$. We assume that doubling condition holds for the metric measure space $X$. We show that a sharp Hörmander-type spectral multiplier theorem on $H^p_L(X)$ follows from restriction type estimates and the Davies-Gaffney estimates. We also describe the sharp result for the boundedness of Bochner-Riesz means on $H^p_L(X)$.
Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. I…
We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_α\ast u^p)u^q$ in $Ω$, where $Ω\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_α$ stands for the Riesz potential of order $α\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $α$ for which positive solutions exist. Our methods are then extended to quasilinear elli…
Weak type $(1,1)$ estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators
Institut de Mathématiques de Bordeaux, Université de Bordeaux, UMR CNRS 5251（波尔多数学研究所，波尔多大学）；International Research Laboratory FAMSI, CNRS-ANU（FAMSI国际研究实验室，法国国家科学研究中心-澳大利亚国立大学）
In this paper, Mikhlin and Marcinkiewicz--Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. By using this results embedding theorems in Sobolev-Lions type spaces is obtained. Moreover, maximal regularity properties of abstract elliptic and parabolic equations are derived
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a …
Second order divergence form operators are studied on an open set with various boundary conditions. It is shown that the p-ellipticity condition of Carbonaro-Dragicevic and Dindos-Pipher implies extrapolation to a holomorphic semigroup on Lebesgue spaces in a p-dependent range of exponents that extends the maximal range for general strictly elliptic coefficients. Results have immediate consequences for the harmonic analysis of such operators, including Hoo-calculi and Riesz transforms.
It is well known that the Stein-Tomas $L^2$ Fourier restriction theorem can be used to derive sharp $L^p$ bounds for radial Fourier multipliers such as the Bochner-Riesz means. In a similar manner, $L^p \to L^2$ estimates for spectral projection operators have been utilized in order to obtain sharp $L^p$ bounds for spectral multipliers of self-adjoint elliptic pseudo-differential operators on compact manifolds. In this paper, we refine an endpoint result for spectral multipliers due to Seeger, providing endpoint bounds in terms of Besov spaces. Our proof is based on the ideas from the recent w…
Let $G = N \rtimes A$, where $N$ is a stratified group and $A = \mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $Δ$ on $G$. Here we prove weak type $(1,1)$, $L^p$-boundedness for $p \in (1,2]$ and $H^1 \to L^1$ boundedness of the Riesz transforms $Y Δ^{-1/2}$ and $Y Δ^{-1} Z$, where $Y$ and $Z$ are any horizontal left-invariant vector fields on $G$, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial deri…
In this paper second order elliptic boundary value problems on bounded domains $Ω\subset\dR^n$ with boundary conditions on $\partialΩ$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space $L^2(Ω…
