Joaquim Martín

Sobolev inequalities: Symmetrization and self-improvement via truncation☆

Abstract We develop a new method to obtain symmetrization inequalities of Sobolev type. Our approach leads to new inequalities and considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces.

POINTWISE SYMMETRIZATION INEQUALITIES FOR SOBOLEV FUNCTIONS AND APPLICATIONS

We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.

Symmetrization inequalities and Sobolev embeddings

We prove new extended forms of the Polya-Szego symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

Abstract Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.

Endpoint estimates from restricted rearrangement inequalities

Let T be a sublinear operator such that (Tf) ∗ (t) ≤ h(t, � f � 1) for some positive function h(t,s) and every function f such that � f � ∞ ≤ 1. Then, we show that T can be extended continuously from a logarithmic type space into a weighted weak Lorentz space. This type of result is connected with the theory of restricted weak type extrapolation and extends a recent result of Arias-de-Reyna concerning the pointwise convergence of Fourier series to a much more general context.

Self Improving Sobolev-Poincaré Inequalities, Truncation and Symmetrization

In Martín et al. (J Funct Anal 252:677–695, 2007) we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in

Equivalent expressions for norms in classical Lorentz spaces

W_{0}^{1,1}(\Omega)

Some reiteration results for interpolation methods defined by means of polygons

. In this paper we extend our method to Sobolev functions that do not vanish at the boundary.

On interpolation of function spaces by methods defined by means of polygons

Abstract We characterize the weights w such that ∫0 ∞ ƒ*(s)p w(s) ds≃ ∫0 ∞ (ƒ**(s) – ƒ*(s))p w(s) ds. Our result generalizes a result due to Bennett–De Vore–Sharpley, where the usual Lorentz L p,q norm is replaced by an equivalent expression involving the functional ƒ ** – ƒ *. Sufficient conditions for the boundedness of maximal Calderón–Zygmund singular integral operators between classical Lorentz spaces are also given.

Weighted norm inequalities and indices

We continue the research on reiteration results between interpolation methods associated to polygons and the. real method. Applications are given to N-tuples of function spaces, or spaces or hounded linear operators and Banach algebras.