Remarks on singular convolution operators
Restrictions from to of weak type (1,1) multipliers
Rough Marcinkiewics integral operators.
Rough Marcinkiewicz integral operators on product spaces.
In this paper, we study the Marcinkiewicz integral operators MΩ,h on the product space Rn x Rm. We prove that MΩ,h is bounded on Lp(Rn x Rm) (1< p < ∞) provided that h is a bounded radial function and Ω is a function in certain block space Bq(0,0) (Sn−1 x Sm−1) for some q > 1. We also establish the optimality of our condition in the sense that the space Bq(0,0) (Sn−1 x Sm−1) cannot be replaced by Bq(0,r) (Sn−1 x Sm−1) for any −1 < r < 0. Our results improve some...
Rough Maximal Oscillatory Singular Integral Operators
2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25In this paper, we establish the L^p boundedness of certain maximal oscillatory singular integral operators with rough kernels belonging to certain block spaces. Our L^p boundedness result improves previously known results.
Rough singular integrals on product spaces.
Sharp -weighted Sobolev inequalities
We prove sharp weighted inequalities of the formwhere is a differential operator and is a combination of maximal type operator related to and to .
Sharp Lp estimates for the segment multiplier.
Sharp spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates
We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let (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 follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on .
Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms
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...
Singular integrals and the Newton diagram.
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.
Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves
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 -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.
Sobolev multipliers
Some multiplier theorems on the sphere.
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...
Some new light on a few classical results
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...
Some notes to certain modification of the Oseen problem
Some spherical uniqueness theorems for multiple trigonometric series.
Spaces of sequences, sampling theorem, and functions of exponential 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.
Spectral decompositions and harmonic analysis on UMD 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 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.
Spectral multipliers and multiple-parameter structures on the Heisenberg group