Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability (original) (raw)
Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means
Journal d'Analyse Mathématique, 2016
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 dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on R n , the harmonic oscillator, elliptic operators on compact manifolds and Schrödinger operators on asymptotically conic manifolds.
Plancherel-type estimates and sharp spectral multipliers
Journal of Functional Analysis, 2002
We study general spectral multiplier theorems for self-adjoint positive definite operators on L 2 (X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L 2 norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R 3 and new spectral multiplier theorems for the Laguerre and Hermite expansions.
Journal of Fourier Analysis and Applications, 2016
Let X 1 and X 2 be metric spaces equipped with doubling measures and let L 1 and L 2 be nonnegative self-adjoint second-order operators acting on L 2 (X 1) and L 2 (X 2) respectively. We study multivariable spectral multipliers F(L 1 , L 2) acting on the Cartesian product of X 1 and X 2. Under the assumptions of the finite propagation speed property and Plancherel or Stein-Tomas restriction type estimates on the operators L 1 and L 2 , we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator F(L 1 , L 2) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space X 1 × X 2. We apply our results to the analysis of secondorder elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner-Riesz means. Contents 1. Introduction 1 2. Notation and preliminary results 6 2.1. Finite propagation speed property for the wave equation 8 2.2. Stein-Tomas restriction type estimates 9 3. The Hardy space H p L 1 ,L 2 (X 1 × X 2) 9 3.1. Atomic decomposition for the Hardy space H 1 L 1 ,L 2 (X 1 × X 2) 10 3.2. An interpolation theorem 12 4. Multivariable spectral multiplier theorems 12 5. Off-diagonal estimates for multivariable spectral multipliers 14 6. Proofs of Theorems 4.1 and 4.2 23 7. Applications 30 7.1. Riesz-transform-like operators 30 7.2. Double Bochner-Riesz means 31 References 31
Partial spectral multipliers and partial Riesz transforms for degenerate operators
Revista Matemática Iberoamericana, 2013
We consider degenerate differential operators of the type A = − d k,j=1 ∂ k (a kj ∂j) on L 2 (R d) with real symmetric bounded measurable coefficients. Given a function χ ∈ C ∞ b (R d) (respectively, a bounded Lipschitz domain Ω), suppose that (a kj) ≥ μ > 0 a.e. on supp χ (respectively, a.e. on Ω). We prove a spectral multiplier type result: if F : [0, ∞) → C is such that sup t>0 ϕ(.)F (t.) C s < ∞ for some nontrivial function ϕ ∈ C ∞ c (0, ∞) and some s > d/2 then MχF (I + A)Mχ is weak type (1, 1) (respectively, PΩF (I + A)PΩ is weak type (1, 1)). We also prove boundedness on L p for all p ∈ (1, 2] of the partial Riesz transforms Mχ∇(I + A) −1/2 Mχ. The proofs are based on a criterion for a singular integral operator to be weak type (1, 1).
Spectral multiplier theorem for H1H^1H1 spaces associated with some Schrödinger operators
Proceedings of the American Mathematical Society, 1999
Let Tt be the semigroup of linear operators generated by a Schrödinger operator −A = ∆ − V , where V is a nonnegative polynomial. We say that f is an element of H 1 A if the maximal function Mf (x) = sup t>0 |Ttf (x)| belongs to L 1. A criterion on functions F which implies boundedness of the operators F (A) on H 1 A is given.
Sharp spectral multipliers for operators satisfying generalized Gaussian estimates
Journal of Functional Analysis, 2013
Let L be a non-negative self adjoint operator acting on L 2 (X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernels p t (x, y) satisfy generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein-Tomas type estimates. These results are applicable to spectral multipliers for large classes of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.
Notes on boundedness of spectral multipliers on Hardy spaces associated to operators
Nagoya Mathematical Journal, 2011
Let L be a nonnegative self-adjoint operator on L2 (X), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e–tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L) is bounded on for 0 < p < 1, the Hardy space associated to operator L, when F is a suitable function.
Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers
Journal of Functional Analysis, 2011
Let L be a non-negative self adjoint operator acting on L 2 (X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernels p t (x, y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted L p -norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L 2 estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.
Weighted estimates in a limited range with applications to the Bochner-Riesz operators
Indiana University Mathematics Journal, 2012
From weighted inequalities for weights in subsets of A p classes, we deduce two kinds of weighted estimates, from L p (u) to L q,∞ (v) and from the weighted Lorentz space Λ p (w) to Λ p,∞ (w). The applications include the Bochner-Riesz operators and some others. We also consider the results in the case of one-sided operators.
On some operator-valued Fourier pseudo-multipliers associated to Grushin operators
arXiv (Cornell University), 2023
This is a continuation of our work [BBGG23, BBGG22] where we have initiated the study of sparse domination and quantitative weighted estimates for Grushin pseudomultipliers. In this article, we further extend this analysis to study analogous estimates for a family of operator-valued Fourier pseudo-multipliers associated to Grushin operators G = −∆ x ′ − |x ′ | 2 ∆ x ′′ on R n1+n2 .
Weighted Hardy spaces associated to operators and boundedness of singular integrals
Arxiv preprint arXiv:1202.2063, 2012
Let (X, d, µ) be a space of homogeneous type, i.e. the measure µ satisfies doubling (volume) property with respect to the balls defined by the metric d. Let L be a non-negative self-adjoint operator on L 2 (X). Assume that the semigroup of L satisfies the Davies-Gaffney estimates. In this paper, we study the weighted Hardy spaces H p L,w (X), 0 < p ≤ 1, associated to the operator L on the space X. We establish the atomic and the molecular characterizations of elements in H p L,w (X). As applications, we obtain the boundedness on H p L,w (X) for the generalized Riesz transforms associated to L and for the spectral multipliers of L. Contents 1. Introduction 2. Preliminaries 2.1. Doubling metric spaces 2.2. Muckenhoupt weights 3. Weighted Hardy spaces associated to operators 3.1. Definition of weighted Hardy spaces 3.2. Finite propagation speed for the wave equation 3.3. Weighted tent spaces 3.4. Atomic characterization of weighted Hardy spaces H p L,w (X) 3.5. Molecular characterization of weighted Hardy spaces H p L,w (X) 4. Boundedness of singular integrals with non-smooth kernels 4.1. Generalized Riesz transforms 4.2. Boundedness of Riesz transforms associated with magnetic Schrödinger operators 4.3. Spectral multiplier theorem on H p L,w (X) References
International Mathematics Research Notices, 2019
Let XXX be a space of homogeneous type and let LLL be a nonnegative self-adjoint operator on L2(X)L^2(X)L2(X) that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for LLL on the Besov and Triebel–Lizorkin spaces associated to LLL. Our work not only recovers the boundedness of the spectral multipliers on LpL^pLp spaces and Hardy spaces associated to LLL but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.
Bulletin des Sciences Mathématiques, 2013
Let T be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of R n. We assume that the associated kernel of T satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for T and the boundedness of the commutator of T with BMO functions on products of weighted Lebesgue spaces of R n. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of their commutators with BMO functions on the products of weighted Lebesgue spaces when the number of derivatives of the symbols is the same as the best known result for the multilinear Fourier multipliers to be bounded on the products of unweighted Lebesgue spaces. Contents
Journal of the Mathematical Society of Japan, 2011
Let (X, d, µ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure µ. Let L be a non-negative self-adjoint operator on L 2 (X). Assume that the semigroup e −tL generated by L satisfies the Davies-Gaffney estimates. Let H p L (X) be the Hardy space associated with L. We prove a Hörmander-type spectral multiplier theorem for L on H p L (X) for 0 < p < ∞: the operator m(L) is bounded from H p L (X) to H p L (X) if the function m possesses s derivatives with suitable bounds and s > n(1/p − 1/2) where n is the "dimension" of X. By interpolation, m(L) is bounded on H p L (X) for all 0 < p < ∞ if m is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on L p spaces with appropriate weights in the reverse Hölder class.
Weighted bounds for multilinear operators with non-smooth kernels
arXiv: Classical Analysis and ODEs, 2015
Let TTT be a multilinear {integral} operator which is bounded on certain products of Lebesgue spaces on mathbbRn\mathbb R^nmathbbRn. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\"older continuity of those in the class of multilinear Calder\'on-Zygmund singular integral operators. In this paper, given a suitable multiple weight vecw\vec{w}vecw, we obtain the bound for the weighted norm of multilinear operators TTT in terms of vecw\vec{w}vecw. As applications, we exploit this result to obtain the weighted bounds {for} certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schr\"odinger operators on mathbbRn\mathbb{R}^nmathbbRn and these results are new in the literature