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1.
Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup etL whose kernels pt(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 Lp-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 L2 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.  相似文献   

2.
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 kernel p t (x,y) has a Gaussian upper bound 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 a weighted Hörmander-type spectral multiplier theorem follows from weighted L p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L 2 estimates of the kernels of the spectral multipliers.  相似文献   

3.
We consider abstract non-negative self-adjoint operators on L2(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 Rn, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.  相似文献   

4.
We study operators \(f\mapsto Kf\) of the form \((Kf)(t)=\int_{{\bf R}^{n}} k(t-s)f(s) {\rm d}s\), where f is a vector-valued function and k an operator-valued singular kernel. Sufficient conditions for boundedness on L p -spaces of UMD-valued functions are given in terms of a Hörmander-type condition involving R-boundedness. The results cover large classes of kernels and also provide new proofs of recent operator-valued Fourier multiplier theorems. Moreover, they give new information about families of singular integral operators.  相似文献   

5.
Let L   be a non-negative self-adjoint operator acting on L2(X)L2(X) where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e−tLetL which satisfies 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 a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.  相似文献   

6.
In [3] and [4]Kitada presented Hörmander-type multiplier theorems for Lebesgue and Hardy spaces defined over a locally compact Vilenkin groupG. Like in the classical case, multipliers for the spaceL 1(G) were not included in these results. In the present paper we discuss this particular case and we show how we need to modify the usual Hörmander multiplier condition to obtainL 1 (G)-multipliers.  相似文献   

7.
In this note we announce L p multiplier theorems for invariant and noninvariant operators on compact Lie groups in the spirit of the well-known Hörmander-Mikhlin theorem on ? n and its versions on the torus $\mathbb{T}^n$ . Applications to mapping properties of pseudo-differential operators on L p -spaces and to a priori estimates for nonhypoelliptic operators are given.  相似文献   

8.
We study Schrödinger operators with Robin boundary conditions on exterior domains in ? d . We prove sharp point-wise estimates for the associated semigroups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.  相似文献   

9.
We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for H?rmander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for H?rmander-type oscillatory integral operators in even dimensions. These last oscillatory estimates are related to improved bounds on the dimensions of curved Kakeya sets in even dimensions.  相似文献   

10.
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.  相似文献   

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