Corrigendum to “Convexity and concavity constants in Lorentz and Marcinkiewicz spaces” [J. Math. Anal. Appl. 343 (2008) 337-351]

In this note we state the accurate formulas of convexity and concavity constants in Lorentz spaces, correcting the formulas from Kamińska and Parrish (2008) [2].     

Orlicz spaces with convexity or concavity constant one

We characterize the classes of function and sequence Orlicz spaces that satisfy upper or lower p-estimate with constant one. We also present a characterization of p-convexity or p-concavity with constant one in Orlicz spaces.     

An off-diagonal Marcinkiewicz interpolation theorem on Lorentz spaces

Let (X, μ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L ^p,q (X) with p, q ∈ (0,∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.: Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ∈ [1,∞) and q ∈ [1,∞), was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].     

Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces

This paper is devoted to the interpolation principle between spaces of weak type. We characterise interpolation spaces between two Marcinkiewicz spaces in terms of Hardy type operators involving suprema. We study general properties of such operators and their behavior on Lorentz gamma spaces. A particular emphasis is placed on elementary and comprehensive proofs.     

Interpolation of bilinear operators in Marcinkiewicz spaces

A theorem on interpolation of bilinear operators in symmetric Marcinkiewicz spaces is proved. It follows from the general bilinear results for the Peetre and Peetre-Gustavsson interpolation functors. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 483–494, October, 1996.     

Banach-Saks property in Marcinkiewicz spaces

We catalogue all Marcinkiewicz function and sequence spaces with the Banach-Saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p-Banach-Saks property, 1<p<∞. We apply our results to several open problems.     

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.     

M‐constants in Orlicz–Lorentz function spaces

In this paper some lower and upper estimates of M‐constants for Orlicz–Lorentz function spaces for both, the Luxemburg and the Amemiya norms, are given. Since degenerated Orlicz functions φ and degenerated weighted sequences ω are also admitted, this investigations concern the most possible wide class of Orlicz–Lorentz function spaces. M‐constants were defined in 1969 by E. A. Lifshits, and used by many authors in the study of lattice structures on Banach spaces, as well as in the fixed point theory.     

Sobolev type spaces in quantum calculus

In this paper q-Sobolev type spaces are defined on Rq by using the q-cosine Fourier transform and its inverse. In particular, embedding results for these spaces are established. Next we define the q-cosine potential and study some of its properties.     

Lorentz spaces with variable exponents

We introduce Lorentz spaces and with variable exponents. We prove several basic properties of these spaces including embeddings and the identity . We also show that these spaces arise through real interpolation between and . Furthermore, we answer in a negative way the question posed in 12 whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.     

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

For the classical Hardy-Littlewood maximal function , a well known and important estimate due to Herz and Stein gives the equivalence . In the present note, we study the validity of analogous estimates for maximal operators of the form

where denotes the Lorentz space -norm.

     

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.     

A note on Marcinkiewicz integral operators

This paper is primarily concerned with proving the L^p boundedness of Marcinkiewicz integral operators with kernels belonging to certain block spaces. We also show the optimality of our condition on the kernel for the L² boundedness of the Marcinkiewicz integral.     

Convexity conditions and normal structure of Banach spaces

We prove that F-convexity, the property dual to P-convexity of Kottman, implies uniform normal structure. Moreover, in the presence of the WORTH property, normal structure follows from a weaker convexity condition than F-convexity. The latter result improves the known fact that every uniformly nonsquare space with the WORTH property has normal structure.     

Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces

In this paper, the necessary and sufficient conditions are found for the boundedness of the rough B-fractional integral operators from the Lorentz spaces Lp,s,γ to Lq,r,γ, 1<p<q<∞, 1?r?s?∞, and from L1,r,γ to Lq,∞,γ≡WLq,γ, 1<q<∞, 1?r?∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.     

Interpolative construction and the generalized cotype of abstract Lorentz spaces

The Banach operator ideals generated by an interpolative construction depending on concave functions are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. The abstract notion of Rademacher type and cotype for operators between Banach spaces is introduced. It is shown that abstract interpolation Lorentz spaces that appeared in the factorization theorem are of the generalized Rademacher cotype determined by Orlicz sequence spaces.     

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.