The bilinear Hilbert trasform is pointwise finite.
Let f ∈ L∞ and g ∈ L2 be supported on [0,1]. Then the principal value integral below exists in L1.p.v. ∫ f(x + y) g(x - y) dy / y.
The class Bpfor weighted generalized Fourier transform inequalities
In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.
The Dunkl transform.
The Ehrhart polynomial of a lattice -simplex.
The Fourier transform in weighted Lorenz spaces.
The Fourier transforms of Lipschitz functions on certain domains.
The fundamental solution for a consistent complex model of the shallow shell equations.
The Stieltjes transform of distributions.
The Uncertainty Principle: A Mathematical Survey.
Three results in Dunkl analysis
We first establish a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the norm of Dunkl translations in dimension 1. Finally, we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.
Trace Inequalities, Maximal Inequalities, and Weighted Fourier Transform Estimates.
Transferring eigenfunction bounds from to hⁿ
By using the notion of contraction of Lie groups, we transfer estimates for joint spectral projectors from the unit complex sphere in to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.
Transformée de Fourier et stabilité sur les surfaces abéliennes
Two problems associated with convex finite type domains.
We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp Lp estimates for p > 4, generalizing the Carleson-Sjölin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.
Two results on the Dunkl maximal operator
In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the case by establishing a result of exponential integrability corresponding to the case p = +∞.