scholarly journals THE CONVOLUTION IN ANISOTROPIC TRIEBEL–LIZORKIN SPACES

2020 ◽  
Vol 69 (1) ◽  
pp. 163-168
Author(s):  
N.T. Tleukhanova ◽  
◽  
K.K. Sadykova ◽  
Keyword(s):  
Harmonic Analysis ◽  
Convolution Operator ◽  
Lorentz Space ◽  
Research Method ◽  
Lorentz Spaces ◽  
Lizorkin Space ◽  
Space Scale ◽  
Anisotropic Case

In this paper, we investigate the boundedness of the norm of the convolution operator in anisotropic Triebel-Lizorkin spaces. The Triebel-Lizorkin spaces are based on the Lorentz spaces pq L . In the anisotropic case, we take the anisotropic Lorentz space pq L as the base. The main goal of the paper is to solve the following problem: let f and g be functions from some classes of the Triebel-Lizorkin space scale. It is necessary to determine which conditions on the parameters of the spaces from f and g are taken and study which space belongs to their convolution gf  . An analogue of the O'Neil theorem was obtained for the Triebel-Lizorkin space scale αq pτF , where α , τ, p , q are vector parameters. Relations of the form γξ hν βη rμ F F  ↪ αq pτF are obtained, with the corresponding ratios of vector parameters γ βα  , hrp 11 1 1   , νμτ 111  , ηξq 111  . The research method is the functional spaces theory and inequalities of functional and harmonic analysis.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

2010 ◽  
Vol 88 (1) ◽  
pp. 19-27 ◽  
Author(s):  
SORINA BARZA ◽  
JAVIER SORIA
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Best Constants ◽  
Sharp Constants ◽  
Dual Norm ◽  
Equivalent Norms ◽  
Weighted Lorentz Spaces ◽  
Weighted Lorentz Space

AbstractFor an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

scholarly journals The uniform Kadec-Klee property for the Lorentz spaces Lw,1

1996 ◽  
Vol 60 (1) ◽  
pp. 7-17 ◽  
Author(s):  
S. J. Dilworth ◽  
Yu-Ping Hsu
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Weak Star

AbstractIn this paper we show that the Lorentz space Lw, 1(0, ∞) has the weak-star uniform Kadec-Klee property if and only if inft>0 (w(αt)/w(t)) > 1 and supt>0(φ(αt) / φ(t))< 1 for all α ∈ (0, 1), where φ(t) = ∫t0 w(s) ds.

Convolution in Weighted Lorentz Spaces of Type $\Gamma$

2016 ◽  
Vol 119 (1) ◽  
pp. 113 ◽  
Author(s):  
Martin Křepela
Keyword(s):  
Convolution Operator ◽  
Lorentz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Weighted Lorentz Spaces

We characterize boundedness of the convolution operator between weighted Lorentz spaces $\Gamma^p(v)$ and $\Gamma^q(w)$ for the range of parameters $p,q\in[1,\infty]$, or $p\in(0,1)$ and $q\in\{1,\infty\}$, or $p=\infty$ and $q\in(0,1)$. We provide Young-type convolution inequalities of the form \[ \|f\ast g\|_{\Gamma^q(w)} \le C \|f\|_{\Gamma^p(v)}\|g\|_Y, \quad f\in\Gamma^p(v), g\in Y, \] characterizing the optimal rearrangement-invariant space $Y$ for which the inequality is satisfied.

scholarly journals Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Khedoudj Saibi
Keyword(s):  
Measurable Function ◽  
Lorentz Space ◽  
Hölder Continuity ◽  
Continuity Condition ◽  
Lorentz Spaces ◽  
Square Function ◽  
Holder Continuity ◽  
Area Function ◽  
Intrinsic Square Function ◽  
Lusin Area Function

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paley g-function, the intrinsic Lusin area function, and the intrinsic gλ∗-function of the variable Hardy–Lorentz space Hp⋅,qℝn, for p⋅ being a measurable function on ℝn satisfying 0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞ and the globally log-Hölder continuity condition and q∈0,∞, via its atomic and Littlewood–Paley function characterizations.

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

2007 ◽  
Vol 143 (2) ◽  
pp. 349-374 ◽  
Author(s):  
A. KAMIŃSKA ◽  
CHRIS LENNARD ◽  
MIECZYSŁAW MASTYŁO ◽  
SYLWIA MIKULSKA
Keyword(s):  
Lorentz Space ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Weak Star

AbstractWe give sufficient conditions, as well as some necessary conditions, for the Orlicz–Lorentz space Λϕ,ω to have the weak-star uniform Kadec–Klee property. These results generalize the characterization of the weak-star uniform Kadec–Klee property in the Lorentz space Λω = Lω,1 due to Dilworth and Hsu.

scholarly journals On boundedness of operators of weak type $(\varphi_0, \psi_0, \varphi_1, \psi_1)$ in Lorentz spaces in limit cases

2021 ◽  
Vol 15 ◽  
pp. 107
Author(s):  
B.I. Peleshenko
Keyword(s):  
Lorentz Space ◽  
Weak Type ◽  
Lorentz Spaces

We prove theorems on boundedness of operators of weak type $(\varphi_0, \psi_0, \varphi_1, \psi_1)$ from Lorentz space $\Lambda_{\varphi,a}(\mathbb{R}^n)$ to $\Lambda_{\varphi,b}(\mathbb{R}^n)$ in “limit” cases, when one of functions $\varphi(t) / \varphi_0(t)$, $\varphi(t) / \varphi_1(t)$ slowly changes at zero and at infinity.

scholarly journals Growth Envelopes of Some Variable and Mixed Function Spaces

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Dorothee D. Haroske ◽  
Cornelia Schneider ◽  
Kristóf Szarvas
Keyword(s):  
Lorentz Space ◽  
Function Spaces ◽  
Lorentz Spaces ◽  
Lebesgue Spaces ◽  
Mixed Norms ◽  
Hardy Type Inequalities ◽  
Variable Lebesgue Spaces ◽  
Hardy Type ◽  
Growth Envelopes ◽  
Natural Way

AbstractWe study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where $$p_{i}$$ p i is the smallest, is crucial. More precisely, the growth envelope is given by $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$ E G ( L p → ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , min { p 1 , … , p d } ) for mixed Lebesgue and $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$ E G ( L p → , q ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , q ) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain $${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$ E G ( L p ( · ) ( Ω ) ) = ( t - 1 / p - , p - ) , where $$p_{-}$$ p - is the essential infimum of $$p(\cdot )$$ p ( · ) , subject to some further assumptions. Similarly, for the variable Lorentz space, it holds$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$ E G ( L p ( · ) , q ( Ω ) ) = ( t - 1 / p - , q ) . The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.

scholarly journals On the solid hull of the Hardy-Lorentz space

2009 ◽  
Vol 85 (99) ◽  
pp. 55-61 ◽  
Author(s):  
Miroljub Jevtic ◽  
Miroslav Pavlovic
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Mixed Norm ◽  
Mixed Norm Space ◽  
Norm Space ◽  
Solid Hull

The solid hulls of the Hardy-Lorentz spaces Hp,q,0 < p < 1, 0 < q ? ? and Hp,? 0, 0 < p < 1, as well as of the mixed norm space H p,?,? 0,0 < p ? 1, 0 < ? < ?, are determined.

Boundedness of integral operators on decreasing functions

2015 ◽  
Vol 145 (4) ◽  
pp. 725-744 ◽  
Author(s):  
María J. Carro ◽  
Carmen Ortiz-Caraballo
Keyword(s):  
Harmonic Analysis ◽  
Weak Type ◽  
Integral Operators ◽  
Lorentz Spaces ◽  
Type Estimate ◽  
Weighted Lorentz Spaces ◽  
Decreasing Functions

We continue the study of the boundedness of the operatoron the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

scholarly journals The Banach-Saks Properties in Orlicz-Lorentz Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Anna Kamińska ◽  
Han Ju Lee
Keyword(s):  
Function Space ◽  
Lorentz Space ◽  
Lorentz Spaces

The Banach-Saks index of an Orlicz-Lorentz spaceΛφ,w(I)for both function and sequence case, is computed with respect to its Matuszewska-Orlicz indices ofφ. It is also shown that an Orlicz-Lorentz function space has weak Banach-Saks (resp., Banach-Saks) property if and only if it is separable (resp., reflexive).

Sign in / Sign up

Export Citation Format

Share Document