La convolution dans faible de
Noyaux singuliers positifs et ensembles exceptionnels
Un contre-exemple pour le noyau reproduisant de la mesure gaussienne dans le plan complexe
Fatou theorems and maximal functions for eigenfunctions of the Laplace-Beltrami operator in a bidisk.
Characterisations of Poisson integrals on symmetric spaces.
On the convergence of bilinear and quadratic forms in independent random variables
Asymptotic behaviour of generalized Poisson integrals in rank one symmetric spaces and in trees
Harmonic spaces associated with adjoints of linear elliptic operators
Let be an elliptic linear operator in a domain in . We imposse only weak regularity conditions on the coefficients. Then the adjoint exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type given distribution. We then apply to R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of are also studied. The results generalize earlier work of the author.
Two-Parameter Maximal Functions in the Heisenberg Group.
Boundary behaviour of eigenfunctions of the Laplacian in a bi-tree.
Singular integrals on Iwasawa NA groups of rank 1.
Singular integrals on the complex affine group
Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1
We study convolution operators bounded on the non-normable Lorentz spaces of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on . In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...
Sharp estimates of the Jacobi heat kernel
The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the other two classical systems of orthogonal polynomials. We deduce sharp estimates giving the order of magnitude of this kernel, for type parameters α, β ≥ -1/2. Using quite different methods, Coulhon, Kerkyacharian and Petrushev recently also obtained such estimates. As an application of the bounds, we show that...
Rough maximal functions and rough singular integral operators applied to integrable radial functions.
Let Ω be homogeneous of degree 0 in R and integrable on the unit sphere. A rough maximal operator is obtained by inserting a factor Ω in the definition of the ordinary maximal function. Rough singular integral operators are given by principal value kernels Ω(y) / |y|, provided that the mean value of Ω vanishes. In an earlier paper, the authors showed that a two-dimensional rough maximal operator is of weak type (1,1) when restricted to radial functions. This result is now extended to arbitrary finite...
Boundedness from to of Riesz transforms on a Lie group of exponential growth
Let be the Lie group endowed with the Riemannian symmetric space structure. Let be a distinguished basis of left-invariant vector fields of the Lie algebra of and define the Laplacian . In this paper we consider the first order Riesz transforms and , for . We prove that the operators , but not the , are bounded from the Hardy space to . We also show that the second-order Riesz transforms are bounded from to , while the transforms and , for , are not.
Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets
Let be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in with respect to the complementary intervals of and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on . Similar properties are studied in for a union of rays from the origin. Then there are connections with the maximal function operator with respect to all rectangles parallel to these rays. In particular, this maximal operator is proved to be bounded on , ,...
Hardy and Lipschitz spaces on subsets of
On the boundary convergence of solutions to the Hermite-Schrödinger equation
In the half-space , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.
Sharp estimates for the Ornstein-Uhlenbeck operator
Let be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure on We prove a sharp estimate of the operator norm of the imaginary powers of on Then we use this estimate to prove that if is in and is a bounded holomorphic function in the sector and satisfies a Hörmander-like condition of (nonintegral) order greater than one on the boundary, then the operator is bounded on This improves earlier results of the authors with J. García-Cuerva...
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