Lorentz-Zygmund Spaces Research Articles

On the spaces introduced by N. K. Karapetyants and B. S. Rubin, and their connection with Lorentz–Zygmund spaces

We consider spaces introduced by N. K. Karapetyants and B. S. Rubin in 1982, to characterize, in particular, the image of the fractional integral Riemann–Liouville operator. These spaces lie near L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{\\infty }$$\\end{document}. We show that they coincide with well-known Lorentz–Zygmund spaces. This allows us to reformulate one result from N. K. Karapetyants and B. S. Rubin dealing with Riemann–Liouville fractional integral operator J0+α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${J}_{0+}^{\\alpha }$$\\end{document} defined on Lp0,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{p}\\left(\ ext{0,1}\\right)$$\\end{document} (1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty$$\\end{document}) in the borderline case α=1/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1/p$$\\end{document}. Using of the well-developed theory of Lorentz–Zygmund spaces leads to new results on the fractional integral Riemann–Liouville operator.

Diversity of Lorentz-Zygmund Spaces of Operators Defined by Approximation Numbers

Diversity of Lorentz-Zygmund Spaces of Operators Defined by Approximation Numbers

Nikol’skii-type inequalities for entire functions of exponential type in Lorentz–Zygmund spaces

Nikol’skii-type inequalities for entire functions of exponential type on \({\mathbb{R}}^{n}\) for the Lorentz–Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov–type spaces of logarithmic smoothness is given.

Nikol’skii–Type Inequalities for Trigonometric Polynomials for Lorentz–Zygmund Spaces

Nikol’skii–type inequalities, that is inequalities between different metrics of trigonometric polynomials on the torus Td for the Lorentz–Zygmund spaces, are obtained. The results of previous paper “Nikol’skii inequalities for Lorentz–Zygmund spaces” are extended. Applications to approximation spaces in Lorentz–Zygmund spaces and to Besov spaces are given.

Associate spaces of logarithmic interpolation spaces and generalized Lorentz–Zygmund spaces

We determine the associate space of the logarithmic interpolation space (X0, X1)1,q,A where X0 and X1 are Banach function spaces over a σ-finite measure space (Ω, µ). Particularizing the results for the case of the couple (L1, L∞) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces L(∞,q;A). We also establish the corresponding results for sequence spaces.

Interpolation between weighted Lorentz spaces with respect to a vector measure

In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties. We describe the interpolation with a parameter function of these spaces. As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund spaces.

LOGARITHMIC INTERPOLATION METHODS AND MEASURE OF NON-COMPACTNESS

Abstract We derive interpolation formulae for the measure of non-compactness of operators interpolated by logarithmic methods with $\theta = 0,1$ between quasi-Banach spaces. Applications are given to operators between Lorentz–Zygmund spaces.

On the generalized Hardy-Rellich inequalities

AbstractIn this paper, we look for the weight functions (sayg) that admit the following generalized Hardy-Rellich type inequality:$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$for some constantC&gt; 0, whereΩis an open set in ℝNwithN⩾ 1. We find various classes of such weight functions, depending on the dimensionNand the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of${\cal D}_0^{2,2} $into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Nikol’skii inequalities for Lorentz–Zygmund spaces

Analogs of the Nikol’skii inequality in Lorentz–Zygmund spaces are obtained for functions of the form $$\sum _{k=1}^{n}c_{k}\varphi _{k}$$ , where $$\left\{ \varphi _{k}\right\} $$ is a finite orthonormal system in $$L_2$$ bounded in $$L_\infty $$ . No assumptions about smoothness of considered functions are required. Used technique relies only on the properties of Lorentz–Zygmund spaces and Fourier series map. In addition, real interpolation method is applied.

Reiteration of a limiting real interpolation method with broken iterated logarithmic functions

A reiteration theorem for a limiting real interpolation method with broken iterated logarithmic functions is established. An application to the generalized Lorentz-Zygmund spaces is given.

Characterization of interpolation between Grand, small or classical Lebesgue spaces

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz–Zygmund spaces or more generally GΓ-spaces. As a direct consequence of our results any Lorentz–Zygmund space La,r(LogL)β, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1<a<∞,β≠0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.

Ul'yanov-type inequalities and embeddings between Besov spaces: the case of parameters with limit values

In this paper we obtain some limit cases of inequalities of Ul'yanov-type for modulus of smoothness between Lorentz-Zygmund spaces on Tn . Corresponding embedding theorems for the Besov spaces are investigated.

Limiting reiteration for real interpolation with logarithmic functions

The real interpolation method \( {\overline{X}}_{\theta ,q,b}=(X_0,X_1)_{\theta ,q,b} \) involving iterated logarithms with any number of iterations is considered. Reiteration relations of the types \( (X_0,{\overline{X}}_{0,q,a})_{\theta ,r,b} \) and \( ({\overline{X}}_{1,q,a}X_1)_{\theta ,r,b} \) \( (0\le \theta \le 1)\) are investigated. Using any number of iterations allows in particular obtaining effects where the resulting space includes three iterated logarithms although the initial scale includes only the uniterated logarithm. Application to Lorentz–Zygmund spaces is given.

On the Relationship Between Two Kinds of Besov Spaces with Smoothness Near Zero and Some Other Applications of Limiting Interpolation

Using limiting interpolation techniques we study the relationship between Besov spaces \(\mathbf B ^{0,-1/q}_{p,q}\) with zero classical smoothness and logarithmic smoothness \(-1/q\) defined by means of differences with similar spaces \(B^{0,b,d}_{p,q}\) defined by means of the Fourier transform. Among other things, we prove that \(\mathbf B ^{0,-1/2}_{2,2}=B^{0,0,1/2}_{2,2}\). We also derive several results on periodic spaces \(\mathbf B ^{0,-1/q}_{p,q}(\mathbb {T})\), including embeddings in generalized Lorentz–Zygmund spaces and the distribution of Fourier coefficients of functions of \(\mathbf B ^{0,-1/q}_{p,q}(\mathbb {T})\).

Fourier Series in Weighted Lorentz Spaces

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces.

Approximation spaces, limiting interpolation and Besov spaces

With the help of limiting interpolation we determine the spaces obtained by iteration of approximation constructions. Then we apply the reiteration formula and limiting interpolation to investigate several problems on Besov spaces, including embeddings in Lorentz–Zygmund spaces and distribution of Fourier coefficients.

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W and the space B (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.

Embeddings of Besov spaces of logarithmic smoothness

This paper deals with Besov spaces of logarithmic smoothness B-p,T(0,b) formed by periodic functions. We study embeddings of B-p,T(0,b) into Lorentz-Zygmund spaces L-p,L-q(log L)(beta). Our techniques rely on the approximation structure of B-p,T(0,b), Nikol'skii type inequalities, extrapolation properties of L-p,L-q(log L)(beta) and interpolation.

Limiting variants of Krasnoselʼskiĭʼs compact interpolation theorem

We illustrate how limiting variants of Krasnoselʼskiĭʼs compact interpolation theorem may be obtained. As a special case of our results it follows that compactness is preserved on certain Lorentz–Zygmund spaces close to the end-point Lebesgue spaces.

A note on Lorentz–Zygmund spaces

Abstract Let 1 ≤ q &lt; p &lt; ∞, and ℝ be the real line. Hörmander showed that any bounded linear translation invariant operator from Lp (ℝ) to Lq (ℝ) is trivial. Blozinski obtained an analogy to Hörmander in Lorentz spaces on the real line. In this paper, we generalize Blozinski's result in Lorentz–Zygmund spaces. Also, Bochkarev proved an inequality related to the Hausdorff–Young–Riesz theorem in Lorentz spaces, and the sharpness of the inequality. We improve Bochkarev's inequality in Lorentz–Zygmund spaces, and prove the sharpness of our inequality.