We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $χ\in C_b^\infty(\mathbb{R}^d)$ (respectively, $Ω$ a bounded Lipschitz domain) and suppose that $(a_{kj}) \ge μ> 0$ a.e.\ on $ \supp χ$ (resp., a.e.\ on $Ω$). We prove a spectral multiplier type result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \| φ(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $φ\in C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_χF(I+A) M_χ$ is weak type $(1,1)$ (resp.\ $P_ΩF(I+A) P_Ω$ is weak type $(1,1)$). We also prove boundedness on $L^p$ for all $p \in (1,2]$ of the partial Riesz transforms $M_χ\nabla (I + A)^{-1/2}M_ χ$. The proofs are based on a criterion for a singular integral operator to be weak type $(1,1)$.
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…
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 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…
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.
The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case
${\mathcal R}$-boundedness of Riesz transforms and spectral multipliers of 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）
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…
In this article, we address sparse bounds for a class of spectral multipliers that include oscillating multipliers on stratified Lie groups. Our results can be applied to obtain weighted bounds for general Riesz means and for solutions of dispersive equations.
We study multivariable spectral multipliers $F(L_1,L_2)$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_1$ and $L_2$. We prove that if $F$ satisfies Hörmander type differentiability condition then the operator $F(L_1,L_2)$ is of Calderón-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product of some fractal spaces. The existence and surprising properties of quasielliptic operators have been recently observed in works of Bockelman, Drenning and Strichartz. This paper demonstrates that Riesz type operators correspondin…
We study multipliers associated to the Hermite operator $H=-Δ+ |x|^2$ on modulation spaces $M^{p,q}(\mathbb R^d)$. We prove that the operator $m(H)$ is bounded on $M^{p,q}(\mathbb R^d)$ under standard conditions on $m,$ for suitable choice of $p$ and $q$. As an application, we point out that the solutions to the free wave and Schrödinger equations associated to $H$ with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to $H$ are bounded on some modulation 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…
Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error e…
We define Hölder classes $Λ_α$ associated with a Markovian semigroup and prove that, when the semigroup satisfies the $Γ^2 \geq 0$ condition, the Riesz transforms are bounded between the Hölder classes. As a consequence, this bound holds in manifolds with nonnegative Ricci curvature. We also show, without the need for extra assumptions on the semigroup, that a version of the Morrey inequalities is equivalent to the ultracontractivity property. This result extends the semigroup approach to the Sobolev inequalities laid by Varopoulos. After that, we study certain families of operators between th…
We prove a sharp multiplier theorem of Mihlin-Hörmander type for the Grushin operator on the unit sphere in $\mathbb{R}^3$, and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.
We study the Grushin operators acting on $\R^{d_1}_{x'}\times \R^{d_2}_{x"}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 - (\sum_{\jone=1}^{d_1}|x'_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x"_\jtwo}^2. \] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These multiplier results are sharp if $d_1 \ge d_2$. We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$. The described spectral multiplier…
The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent $p$ and regularity $s$. Counterexamples are included to show that these results are indeed sharp. In particu…
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 $H^p_L(X)$.
We develop a new approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by Stinga and Torrea, which is of independent interest. Our methods here also recover the sharp $L^p$ bounds for second order Riesz transforms by a liming argument.
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…
The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic se…
This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schrödinger operators $H=-Δ+V$, i.e., In $\mathbb{R}^n$ ($n\ge 3$) we prove the $ε-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-λ\pm iε)^{-1}$ for all $λ>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schrödinger operators $H$. Next, we consider the fractional Sch…
Let $A = - \sum \partial_k \, c_{kl} \, \partial_l$ be a degenerate sectorial differential operator with complex bounded mesaurable coefficients. Let $Ω\subset \mathds{R}^d$ be open and suppose that $A$ is strongly elliptic on $Ω$. Further, let $χ\in C_{\rm b}^\infty(\mathds{R}^d)$ be such that an $\varepsilon$-neighbourhood of $\supp χ$ is contained in $Ω$. Let $ν\in (0,1]$ and suppose that the ${c_{kl}}_{|Ω} \in C^{0,ν}(Ω)$. Then we prove (Hölder) Gaussian kernel bounds for the kernel of the operator $u \mapsto χ\, S_t (χ\, u)$, where $S$ is the semigroup generated by $-A$. Moreover, if $ν= …
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.
The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(λ)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $λ^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setti…
In this paper we prove a Wiener-type characterization of boundary regularity, in the spirit of a classical result by Landis, for a class of evolutive Hörmander operators. We actually show the validity of our criterion for a larger class of degenerate-parabolic operators with a fundamental solution satisfying suitable two-sided Gaussian bounds. Our condition is expressed in terms of a series of balayages or, (as it turns out to be) equivalently, Riesz-potentials.
In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated to the biharmonic operator $Δ^2.$ With this characterizations, we prove the boundedness of the Bessel potentials, fractional integrals, fractional powers, Riesz transforms and multipliers of Laplace transforms type associated to $Δ^2$ on Lipschitz spaces and homogeneous Lipschitz spaces. The proofs of these results need the language of semigroups in an essential way.
In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),γ_α)$ to $L^1((0,\infty),γ_α)$ and also from $L^\infty((0,\infty),γ_α)$ to $\BMO((0,\infty),γ_α)$, with respect to the probability measure $dγ_α(x)=\frac{2}{Γ(α+1)} x^{2α+1} e^{-x^2} dx$ on $(0,\infty)$ when ${α>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.
We derive Hölder regularity estimates for operators associated with a time independent Schrödinger operator of the form $-Δ+V$. The results are obtained by checking a certain condition on the function $T1$. Our general method applies to get regularity estimates for maximal operators and square functions of the heat and Poisson semigroups, for Laplace transform type multipliers and also for Riesz transforms and negative powers $(-Δ+V)^{-γ/2}$, all of them in a unified way.
We study a class of generalized Laplacian operators by violating the ellipticity with degenerate metric tensors. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the metric tensor is given by $g=-\mathcal{J}^2$ with a generalized Poisson matrix $\mathcal{J}$ that dictates particle dynamics. The standard Euclidean metric corresponds to the symplectic matrix of canonical Hamiltonian systems. However, topological constraints bring about nullity to $\mathcal{J}$, resulting in degeneracy in the corresponding diffusion o…
