Gustavo Garrigós

Wavelet Decompositions of Anisotropic Besov Spaces

In this paper we develop the natural multiresolution analysis framework related to anisotropic Besov spaces B p;q (R n). We prove two new Jackson and Bernstein type inequalities for these spaces, and obtain from well-known techniques 12, 7] new norm equivalences in terms of weighted sums of the wavelet coeecients. This provides a characterization for B p;q (R n) (and for its dual) by means of compactly supported wavelets, which may be applied to the numerical resolution of semi-elliptic diierential equations.

On plate decompositions of cone multipliers

An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for a variety of problems in harmonic analysis. We observe that the range in Wolffs inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L^4 bound for the cone multiplier operator in three dimensions.

Wavelet characterizations for anisotropic Besov spaces with 0 p 1

We present a wavelet characterization of anisotropic Besov spaces

Analytic Besov spaces and Hardy-type inequalities in tube domains over symmetric cones

B_{p,q}^{\bm{\alpha}}(\mathbb{R}^n)

Completeness in the set of wavelets

, valid for the whole range

Democracy functions and optimal embeddings for approximation spaces

0\ltp,q\lt\infty

Further results on the connectivity of Parseval frame wavelets

, and in terms of multi-resolution analyses with dilation adapted to the anisotropy of the space. Our proofs combine classical techniques based on Bernstein and Jackson-type inequalities, and nonlinear methods for the cases

Lebesgue inequalities for the greedy algorithm in general bases

p\lt1

Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

. Among the consequences of our results, we characterize

A weak 2-weight problem for the Poisson-Hermite semigroup

B_{p,q}^{\bm{\alpha}}