Multidimensional rearrangement and Lorentz spaces

We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

New Lorentz spaces for the restricted weak-type Hardy's inequalities

Associated to the class of restricted weak-type weights for the Hardy operator R_p, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function.     

Integrability of maximal functions for generalized Lebesgue spaces with variable exponent

Our aim in this paper is to deal with integrability of maximal functions for generalized Lebesgue spaces with variable exponent. Our exponent approaches 1 on some part of the domain, and hence the integrability depends on the shape of that part and the speed of the exponent approaching 1. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardy?s inequality in a limiting case are also considered.     

The dual of noncommutative Lorentz spaces

We prove some noncommutative analogues of the classical results about dual spaces of the classical Lorentz spaces.     

Equivalent quasi-norms on Lorentz spaces

We give new characterizations of Lorentz spaces by means of certain quasi-norms which are shown to be equivalent to the classical ones.

     

Interpolation theorem on Lorentz spaces over weighted measure spaces

In 1997 Ferreyra proved that it is impossible to extend the Stein-Weiss theorem in the context of Lorentz spaces. In this paper we obtain an interpolation theorem on Lorentz spaces over weighted measure spaces.

     

Bilinear multipliers on Lorentz spaces

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform. The author has been partially supported by grants DGESIC PB98-1246 and BMF 2002-04013.     

Generalizations of the Riesz convergence theorem for Lorentz spaces

Summary We obtain preservation inequalities for Lipschitz constants of higher order in simultaneous approximation processes by Bernstein type operators. From such inequalities we derive the preservation of the corresponding Lipschitz spaces.     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.     

Dual of two dimensional Lorentz sequence spaces

In [M. Kato and L. Maligranda, On James and Jordan–von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457–465], it is an open problem to compute the James constant of the dual space of two dimensional Lorentz sequence space d⁽²⁾(w,q). In this paper, we shall determine the dual norm of d⁽²⁾(w,q) and completely compute the James constant of d⁽²⁾(w,q).     

Rearrangement invariance of Rademacher multiplicator spaces

Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of all measurable functions x such that x⋅h∈X for every a.e. converging series h=∑anrn∈X, where (rn) are the Rademacher functions. We study the situation when Λ(R,X) is a rearrangement invariant space different from L^∞. Particular attention is given to the case when X is an interpolation space between the Lorentz space Λ(φ) and the Marcinkiewicz space M(φ). Consequences are derived regarding the behaviour of partial sums and tails of Rademacher series in function spaces.     

Continuity of the maximal operator in Sobolev spaces

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces , . As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

     

Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions

We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the -functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.

     

On the extremality of maximal dual feasible functions

Dual feasible functions have been used to compute bounds and valid inequalities for combinatorial optimization problems. Here, we analyze the properties of some of the best functions proposed so far. Additionally, we provide new results for composed functions. These results will allow improving the computation of bounds and valid inequalities.     

Theoretical investigations on maximal dual feasible functions

Dual feasible functions are used to get valid inequalities and lower bounds for integer linear problems. In this paper, we provide a simpler proof for maximality, and we describe new results concerning the extremality of functions from the literature. Extremal functions are a dominant class of dual feasible functions.     

Level-set inequalities on fractional maximal distribution functions and applications to regularity theory

The aim of this paper is to establish an abstract theory based on the so-called fractional-maximal distribution functions (FMDs). From the basic ideas introduced in [1], we develop and prove some abstract results related to the level-set inequalities and norm-comparisons by using the language of such FMDs. Particularly interesting is the applicability of our approach that has been shown in regularity and Calderón-Zygmund type estimates. In this paper, due to our research experience, we will establish global regularity estimates for two types of general quasilinear problems (problems with divergence form and double obstacles), via fractional-maximal operators and FMDs. The range of applications of these abstract results is large. Apart from these two examples of the regularity theory for elliptic equations discussed, it is also promising to indicate further possible applications of our approach for other special topics.     

Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces

We prove that when a function on the real line is symmetricallyrearranged, the distribution function of its uncentered Hardy–Littlewoodmaximal function increases pointwise, while it remains unchangedonly when the function is already symmetric. Equivalently, if is the maximal operator and the symmetrization, then f(x)f(x)for every x, and equality holds for all x if and only if, upto translations, f(x) = f(x) almost everywhere. Using theseresults, we then compute the exact norms of the maximal operatoracting on Lorentz and Marcinkiewicz spaces, and we determineextremal functions that realize these norms.