Singular integrals on ax+b hypergroups and an operator-valued spectral multiplier theorem

Plancherel-type estimates and sharp spectral multipliers

Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

Spectral multipliers without semigroup framework and application to random walks

Let $(X,d,\mu)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. 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 $H_L^p(X)$ be the Hardy space associated with $L$. We prove a Hormander-type spectral multiplier theorem for $L$ on $H_L^p(X)$ for $0 n(1/p - 1/2)$ where $n$ is the dimension of $X$. By interpolation, $m(L)$ is bounded on $H_L^p(X)$ for all $0 < p < \infty$ 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 Holder class.

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 theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by D. M\uller and E.M. Stein and by W. Hebisch.

An abstract theorem concerning exact sequences of Banach algebras of operators and symbol homomorphisms relative to groups of operators is derived. This general result is used to deduce many of the classical spectral inclusion theorems and short exact sequences for algebras of singular integral operators.

The classical Stein–Tomas restriction theorem is equivalent to the fact that the spectral measure d“E(λ) of the square root of the Laplacian on ℝn is bounded from Lp(ℝn) to Lp′(ℝn) for 1≤p≤2(n+1)∕(n+3), where p′ is the conjugate exponent to p, with operator norm scaling as λn(1∕p−1∕p′)−1. We prove a geometric, or variable coefficient, generalization in which the Laplacian on ℝn is replaced by the Laplacian, plus a suitable potential, on a nontrapping asymptotically conic manifold. It is closely related to Sogge’s discrete L2 restriction theorem, which is an O(λn(1∕p−1∕p′)−1) estimate on the Lp→Lp′ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner–Riesz summability results, which are sharp for p in the range above. The paper divides naturally into two parts. In the first part, we show at an abstract level that restriction estimates imply spectral multiplier estimates, and are implied by certain pointwise bounds on the Schwartz kernel of λ-derivatives of the spectral measure. In the second part, we prove such pointwise estimates for the spectral measure of the square root of Laplace-type operators on asymptotically conic manifolds. These are valid for all λ>0 if the asymptotically conic manifold is nontrapping, and for small λ in general. We also observe that Sogge’s estimate on spectral projections is valid for any complete manifold with C∞ bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on d“E(λ) may blow up exponentially as λ→∞ when trapping is present.

The aim of this paper is to establish a sufficient condition for certain weighted norm inequalities for singular integral operators with non-smooth kernels and for the commutators of these singular integrals with BMO functions. Our condition is applicable to various singular integral operators, such as the second derivatives of Green operators associated with Dirichlet and Neumann problems on convex domains, the spectral multipliers of non-negative self-adjoint operators with Gaussian upper bounds, and the Riesz transforms associated with magnetic Schrodinger operators.

Spectral multipliers from [formula omitted] to [formula omitted

Let M be a measure space and let L be a positive definite operator on L2(M). By the spectral theorem, for any bounded Borel measurable function F : [0, ∞) 7→ C the operator F (L)f = ∫ ∞ 0 F (λ)dE(λ)f is bounded on L2(M). We are interested in sufficient conditions on F for F (L) to be bounded on Lp(M), p 6= 2. We direct the reader to [1], [3], [4], [8], [9], [10], [12] and [13] for more background on various multiplier theorems. In this paper we assume F is compactly supported and have some smoothness (finite number of derivatives) and we consider only the case p = 1. Our measure space G is semidirect product of stratified nilpotent Lie group N and the real line. The operator L is (minus) sublaplacian on G. Our group has exponential volume growth. The earlier theory suggested that one needs holomorphic F for F (L) to be bounded on L1, however the recent results [5], [6], [7] showed that estimates on only a finite number of derivatives of F imply boundedness of F (L) on L1 on some solvable G of exponential growth. In this case we say that G (more precisely L) has Ckfunctional calculus. On the other hand, Christ and Muller give an example of a solvable Lie group on which F must be holomorphic. The problem is to find the condition on G (and possibly L) which decides whether G has a Ck-functional calculus or not. Here, our condition is in terms of roots of adjoint representation of the Lie algebra of G. Our groups are of “rank one”,

Calculating the spectral factorization and outer functions by sampling-based approximations—Fundamental limitations

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0 < p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace--Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schr\"odinger operator on a Euclidean space, and (iii) $T = g(L) $ is a singular integral operator defined from the holomorphic functional calculus of an operator $L$ or the spectral multiplier of a non-negative self-adjoint operator $L$.

We study the L 2 → L ∞ L^2 \to L^{\infty } norms of spectral projectors and spectral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the L 2 → L ∞ L^2 \to L^{\infty } norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos’ spectral multiplier theorem for this class of operators.

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 Hormander-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.