We prove sharp weighted inequalities of the form$${\int }_{{\mathbf{R}}^n}|f\left(x\right){|}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}|q\left(D\right)\left(f\right)\left(x\right){|}^{p}N\left(v\right)\left(x\right)dx$$where $q\left(D\right)$ is a differential operator and $N$ is a combination of maximal type operator related to $q\left(D\right)$ and to $p$.

We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let $H_L^p\left(X\right)$ (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on $H_L^p\left(X\right)$ follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on $H_L^p\left(X\right)$.

We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures...

We examine several scalar oscillatory singular integrals involving a real-analytic phase function φ(s,t) of two real variables and illustrate how one can use the Newton diagram of φ to efficiently analyse these objects. We use these results to bound certain singular integral operators.

The paper presents a theory of Fourier transforms of bounded holomorphic functions defined in sectors. The theory is then used to study singular integral operators on star-shaped Lipschitz curves, which extends the result of Coifman-McIntosh-Meyer on the $L^2$-boundedness of the Cauchy integral operator on Lipschitz curves. The operator theory has a counterpart in Fourier multiplier theory, as well as a counterpart in functional calculus of the differential operator 1/i d/dz on the curves.

The n-dimensional sphere, E, can be seen as the quotient between the group of rotations of R n+1 and the subgroup of all the rotations that fix one point. Using representation theory, one can see that any operator on Lp (Sigma n) that commutes with the action of the group of rotations (called multiplier) may be associated with a sequence of complex numbers. We prove that, if a certain discrete derivative of a given sequence represents a bounded multiplier on LP (E 1), then the given sequence represents...

The purpose of this paper is to describe a unified approach to proving vector-valued inequalities without relying on the full strength of weighted theory. Our applications include the Fefferman-Stein and Córdoba-Fefferman inequalities, as well as the vector-valued Carleson operator. Using this approach we also produce a proof of the boundedness of the classical bi-parameter multiplier operators, which does not rely on product theory. Our arguments are inspired by the vector-valued restricted type...

We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.

We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_X^p$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.

We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^1\left(AN\right),$ but also for $L^p\left(AN\right)$, where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.

We study the $L^p$-boundedness of linear and bilinear multipliers for the symmetric Bessel transform.