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Gagliardo-Nirenberg Inequality for rearrangement-invariant banach function spaces

RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI 2019年第4期30卷 847-864页

作者： Fiorenza, Alberto Formica, Maria Rosaria Roskovec, Tomas Soudsky, Filip Univ Napoli Federico II Dipartimento Architettura Via Monteoliveto 3 I-80134 Naples Italy CNR Ist Applicaz Calcolo Mauro Picone Via Pietro Castellino 11 I-80131 Naples Italy Univ Napoli Parthenope Via Gen Parisi 13 I-80132 Naples Italy Univ South Bohemia Fac Econ Studentska 13 Ceske Budejovice Czech Republic Czech Tech Univ Fac Informat Technol Thakurova 9 Prague 16000 6 Czech Republic

The classical Gagliardo-Nirenberg interpolation inequality is a well-known estimate which gives, in particular, an estimate for the Lebesgue norm of intermediate derivatives of functions in Sobolev spaces. We present an extension of this estimate into the scale of the general rearrangement-invariant banach function spaces with the proof based on the Maz'ya's pointwise estimates. As corollaries, we present the Gagliardo-Nirenberg inequality for intermediate derivatives in the case of triples of Orlicz spaces and triples of Lorentz spaces. Finally, we promote the scaling argument to validate the optimality of the Gagliardo-Nirenberg inequality and show that the presented estimate in Orlicz scale is optimal.

关键词： Gagliardo-Nirenberg inequality banach function spaces Lorentz spaces Orlicz spaces

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Product factorability of integral bilinear operators on banach function spaces

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POSITIVITY 2019年第3期23卷 671-696页

作者： Erdogan, E. Sanchez Perez, E. A. Gok, O. Univ Marmara Fac Art & Sci Dept Math TR-34722 Istanbul Turkey Univ Politecn Valencia Inst Univ Matemat Pura & Aplicada Camino Vera S-N E-46022 Valencia Spain Yildiz Tech Univ Fac Art & Sci Dept Math TR-34349 Istanbul Turkey

This paper deals with bilinear operators acting in pairs of banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices-orthosymmetric maps, C-algebras-zero product preserving operators, and classical and harmonic analysis-integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.

关键词： Integral bilinear operators banach function spaces Factorization Zero product preserving Symmetric operator 47H60 46E30 47A68 47B38

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Polynomial stability of evolution cocycles and banach function spaces

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BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN 2019年第2期26卷 299-314页

作者： Pham Viet Hai Nanyang Technol Univ Sch Phys & Math Sci Div Math Sci Singapore 637371 Singapore

In this paper, we give characterizations for a polynomial stability in banach spaces. This is done by using evolution cocycles and techniques of banach function spaces. Our characterizations are new versions of the th... 详细信息

关键词： polynomial stability evolution cocycles banach function spaces

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An Alternative Approach to Weighted Non-commutative banach function spaces

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COMPLEX ANALYSIS AND OPERATOR THEORY 2019年第7期13卷 3207-3218页

作者： Steyn, Claud NWU Sch Math & Stat Sci DST NRF CoE Math & Stat Sci Unit BMI Internal Box 209Pvt Bag X6001 ZA-2520 Potchefstroom South Africa

We use a weighted analogue of a trace to define a weighted non-commutative decreasing rearrangement and show its relationship with the singular value function. We further show an alternative approach to constructing weighted non-commutative banach function spaces using weighted non-commutative decreasing rearrangements and prove that this approach is equivalent to the original approach by Labuschagne and Majewski.

关键词： Weighted Operator algebras Decreasing rearrangement banach function spaces Measurable operators

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Duality and distance formulas in banach function spaces

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JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS 2019年第1期5卷 1-23页

作者： D'Onofrio, Luigi Sbordone, Carlo Schiattarella, Roberta Univ Napoli Parthenope Naples Italy Univ Napoli Federico II Naples Italy

We consider pairs of non reflexive banach spaces (E-0, E) such that E-0 is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair (E-0, E) there exists a reflexive and separable banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators T : E-0 -> Z where Z is an arbitrary banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, (B-0, B) where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs (L-0(psi), L-psi) where L-0(psi) is the closure of L-infinity in the Orlicz space L-psi, Marcinkiewicz pairs (L-0(q,infinity), L-q,L-infinity) where L-0(q infinity), is the closure of L-infinity in the Marcinkiewicz weak-L-q denoted by L-q,L-infinity. More generally, banach function spaces are considered. The main results are duality formulas of the type E-0** similar or equal to E isometrically (1) E* similar or equal to E-0* circle plus(1) E-0(perpendicular to) (2) and distance formulas.

关键词： banach function spaces Duality Distance formulas

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Bochner representable operators on banach function spaces

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POSITIVITY 2018年第5期22卷 1303-1309页

作者： Nowak, Marian Univ Zielona Gora Fac Math Comp Sci & Econometr Ul Szafrana 4A PL-65516 Zielona Gora Poland

Let be a banach function space, E the Kothe dual of E and (X, center dot X) be a banach space. It is shown that every Bochner representable operator T : E -> X maps relatively sigma (E, E')-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm on E is order continuous, then every Bochner representable operator T : E X is X compact, where E stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.

关键词： banach function spaces Orlicz spaces Mixed topologies Bochner representable operators Compact operators

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Maximal ergodic inequalities for banach function spaces

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INDAGATIONES MATHEMATICAE-NEW SERIES 2016年第1期27卷 29-74页

作者： de Beer, Richard Labuschagne, Louis NWU DST NRF CoE Math & Stat Sci Unit BMI Sch Comp Stat & Math Sci Internal Box 209Pvt Bag X6001 ZA-2520 Potchefstroom South Africa

We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of a-compact locally compact Hausdorff groups acting measure-preservingly on sigma-finite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on different banach function spaces, and how the properties of these function spaces influence the weak type inequalities that can be obtained. Next we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. In closing we briefly indicate the utility of these results for Statistical Physics. (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

关键词： Transfer principle Maximal inequalities banach function spaces Pointwise ergodic theorems

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Some s-numbers of an integral operator of Hardy type in banach function spaces

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JOURNAL OF APPROXIMATION THEORY 2016年 207卷 76-97页

作者： Edmunds, David Gogatishvili, Amiran Kopaliani, Tengiz Samashvili, Nino Univ Sussex Dept Math Pevensey 2North South Rd Brighton BN1 9QH E Sussex England Acad Sci Czech Republ Inst Math Zitna 25 CR-11567 Prague 1 Czech Republic I Javakhishvili Tbilisi State Univ Fac Exact & Nat Sci Univ St 2 GE-0143 Tbilisi Georgia

Let s(n)(T) denote the nth approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator T given by T f(x) = v(x) integral(x)(a) u(t) f(t)dt, x is an element of (a, b) (-infinity < a < b < +infinity) and mapping a banach function space E to itself. We investigate some geometrical properties of E for which C-1 integral(b)(a) u(x)v(x)dx <= lim(n ->infinity) inf ns(n) (T) <= lim(n ->infinity)sup ns(n)(T) <= C-2 integral(b)(a) u(x)v(x)dx under appropriate conditions on it and v. The constants C-1, C-2 > 0 depend only on the space E. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Hardy type operators banach function spaces s-numbers Compact linear operators

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Extending and factorizing bounded bilinear maps defined on order continuous banach function spaces

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2014年第2期108卷 353-367页

作者： Calabuig, J. M. Fernandez Unzueta, M. Galaz-Fontes, F. Sanchez-Perez, E. A. Univ Politecn Valencia Inst Univ Matemat Pura & Aplicada Valencia 46022 Spain Ctr Invest Matemat Guanajuato 36240 Gto Mexico

We consider the problem of extending or factorizing a bounded bilinear map defined on a couple of order continuous banach function spaces to its optimal domain, i.e. the biggest couple of banach function spaces to which the bilinear map can be extended. As in the case of linear operators, we use vector measure techniques to find this space, and we show that this procedure cannot be always successfully used for bilinear maps. We also present some applications to find optimal factorizations of linear operators between banach function spaces.

关键词： Order continuous banach function spaces Vector measures Integrable functions Optimal domain Bilinear map

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Pointwise products of some banach function spaces and factorization

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JOURNAL OF functionAL ANALYSIS 2014年第2期266卷 616-659页

作者： Kolwicz, Pawel Lesnik, Karol Maligranda, Lech Poznan Univ Tech Inst Math Elect Fac PL-60965 Poznan Poland Lulea Univ Technol Dept Engn Sci & Math SE-97187 Lulea Sweden

The well-known factorization theorem of Lozanovskii may be written in the form L-1 E circle dot E', where circle dot means the pointwise product of banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E, i.e., when F E circle dot M(E, F), where M(E, F) is the space of pointwise multipliers from E to F. Properties of M(E, F) were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E circle dot F. The formulas for pointwise product of Calderon-Lozanovskii E-phi-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant banach function space may be factorized through a Marcinkiewicz space. (C) 2013 Elsevier Inc. All rights reserved.

关键词： banach ideal spaces banach function spaces Calderon spaces Calderon-Lozanovskii spaces Symmetric spaces Orlicz spaces Sequence spaces Pointwise multipliers Pointwise multiplication Factorization

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