In this paper we consider a systematic study of several new Lp-boundedness properties for the index 2F1-transform over spaces L?,p (R+), 1 ? p < ?, R, and L? (R+). We also obtain Parseval-type relations these spaces.

Lp−Lq–boundedness of the map f → w(x) ∫ b(x) a(x) k(x, y)f(y)v(y)dy is described by different types of criteria expressed in terms of given parameters 0 < p, q < ∞, strictly increasing boundaries a(x) and b(x), locally integrable weight functions v, w and a positive continuous kernel k(x, y) satisfying some growth conditions.

For the family of truncations of the gaussian Riesz transforms and Poisson integral we study their rate of convergence through the oscillation and variation operators. More precisely, we search for their Lp(dγ)−boundedness properties, being dγ the Gauss measure. We achieve our results by looking at the oscillation and variation operators from a vector valued point of view.

Conditions for boundedness and compactness of product-convolution operators g —» PhCß = h ■ (/» g) on spaces L^G) are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces L,p>q){G) interpolating the Lp(G) spaces in a natural way (L^^ = Z^,). It is then natural to study the operators acting between L(/1?)(G) spaces, where G has a compact invariant neighborhood. The t...

We obtain a maximal transference theorem that relates almost everywhere convergence of multilinear Fourier series to boundedness of maximal multilinear operators. We use this and other recent results on transference and multilinear operators to deduce Lp and almost everywhere summability of certain m-linear Fourier series. We formulate conditions for the convergence of multilinear series and we...