Let X Y and Z be banach function spaces over a measure space . Consider the spaces of multiplication operators from X into the Kothe dual Y' of Y, and the spaces X (Z) and defined in the same way. In this paper we...
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Let X Y and Z be banach function spaces over a measure space . Consider the spaces of multiplication operators from X into the Kothe dual Y' of Y, and the spaces X (Z) and defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey-Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.
We give a sharp sufficient condition on the distribution function, |{x is an element of Q : p ( x ) 0, of the exponent function p () : Q -> [1, infinity) that implies the embedding of the variable Lebesgue space L...
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We give a sharp sufficient condition on the distribution function, |{x is an element of Q : p ( x ) <= 1 + lambda }|, lambda > 0, of the exponent function p () : Q -> [1, infinity) that implies the embedding of the variable Lebesgue space L p ( ) (Q) into the Orlicz space L (log L ) alpha (Q), alpha > 0, where Q is an open set with finite Lebesgue measure. As applications of our results, we first give conditions that imply the strong differentiation of integrals of functions in L p ( ) ((0 , 1)n), n > 1. We then consider the integrability of the maximal function on variable Lebesgue spaces, where the exponent function p () approaches 1 in value on some part of the domain. This result is an improvement of the result in [6]. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper aims to extend several well-known theorems regarding porosity in Lebesgue and Orlicz spaces to generalized Orlicz spaces associated with quasi-banachspaces. We will explore the implications of these extens...
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This paper aims to extend several well-known theorems regarding porosity in Lebesgue and Orlicz spaces to generalized Orlicz spaces associated with quasi-banachspaces. We will explore the implications of these extensions and provide a comprehensive analysis of their significance in the broader context of functional analysis. Additionally, we will present results concerning homogeneous spaces, and relationships with the generalized Orlicz spaces.
作者：
Labuschagne, LouisNWU
DSI NRF CoE Math & Stat Sci Pure & Appl Analyt Sch Math & Stat Sci Internal Box 209PVT BAG X6001 ZA-2520 Potchefstroom South Africa
We revisit the generalisation of Calderon's Transfer Principle as espoused in [7]. This principle is used to generate weak type maximal inequalities for ergodic operators in the setting of sigma-compact locally co...
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We revisit the generalisation of Calderon's Transfer Principle as espoused in [7]. This principle is used to generate weak type maximal inequalities for ergodic operators in the setting of sigma-compact locally compact Hausdorff groups acting measure-preservingly on sigma-finite measure spaces. In particular we develop a much more robust protocol for transferring weak and strong type inequalities from Orlicz spaces in the group setting to Orlicz spaces in the measure space setting. This is an important addition to the protocol developed in [7], which to date has only yielded weak type inequalities. The current approach also places fewer restrictions on the underlying Young functions describing the Orlicz spaces involved.
This paper is devoted to studying the extrapolation theory of Rubio de Francia on general functionspaces. We present endpoint extrapolation results including A1$A_1$, Ap$A_p$, and A infinity$A_\infty$ extrapolation i...
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This paper is devoted to studying the extrapolation theory of Rubio de Francia on general functionspaces. We present endpoint extrapolation results including A1$A_1$, Ap$A_p$, and A infinity$A_\infty$ extrapolation in the context of banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman-Fefferman inequalities that can be used to show some known sharp A1$A_1$ inequalities, Muckenhoupt-Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calderon-Zygmund operators. Finally, we obtain two-weight inequalities for Littlewood-Paley operators and Fourier integral operators on weighted banach function spaces.
Weighted extrapolation for pairs of functions in mixed-norm banach function spaces defined on the product of quasi-metric measure spaces (X,d,& mu;)$(X, d, \mu )$ and (Y,& rho;,& nu;)$(Y, \rho , \nu )$ are...
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Weighted extrapolation for pairs of functions in mixed-norm banach function spaces defined on the product of quasi-metric measure spaces (X,d,& mu;)$(X, d, \mu )$ and (Y,& rho;,& nu;)$(Y, \rho , \nu )$ are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.
We consider m-th order linear, uniformly elliptic equations Lu=f with non-smooth coefficients in banach–Sobolev spaces WXwm(Ω) generated by weighted banach function spaces (BFS) Xw(Ω) on a bounded domain Ω⊂Rn. Sup...
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We consider m-th order linear, uniformly elliptic equations Lu=f with non-smooth coefficients in banach–Sobolev spaces WXwm(Ω) generated by weighted banach function spaces (BFS) Xw(Ω) on a bounded domain Ω⊂Rn. Supposing boundedness of the Hardy–Littlewood Maximal operator and the Calderón–Zygmund singular integrals in Xw(Ω) we obtain solvability in the small in WXwm(Ω) and establish interior Schauder type a priori estimates. These results will be used in order to obtain Fredholmness of the operator L in Xw(Ω).
We present authors' recent results regarding Rubio de Francia's extrapolation, generally speaking, in some mixed norm functionspaces. We study both diagonal and off-diagonal cases. As a consequence, we have t...
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We present authors' recent results regarding Rubio de Francia's extrapolation, generally speaking, in some mixed norm functionspaces. We study both diagonal and off-diagonal cases. As a consequence, we have the boundedness of operators of Harmonic Analysis in these spaces. Some of the properties of mixed norm functionspaces are investigated as well. Some of the presented statements are proved for the first time here.
This paper studies topological duals of locally convex functionspaces that are natural generalizations of Frechet and banach function spaces. The dual is identified with the direct sum of another function space, a sp...
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This paper studies topological duals of locally convex functionspaces that are natural generalizations of Frechet and banach function spaces. The dual is identified with the direct sum of another function space, a space of purely finitely additive measures and the annihilator of L-infinity. This allows for quick proofs of various classical as well as new duality results e.g. in Lebesgue, Musielak-Orlicz, Orlicz-Lorentz space and spaces associated with convex risk measures. Beyond banach and Frdchet spaces, we obtain completeness and duality results in general paired spaces of random variables.
This paper deals with multilinear operators acting in products of banachspaces that factor through a canonical mapping. We prove some factorization theorems and characterizations by means of norm inequalities for mul...
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This paper deals with multilinear operators acting in products of banachspaces that factor through a canonical mapping. We prove some factorization theorems and characterizations by means of norm inequalities for multilinear operators defined on the n-fold Cartesian product of the space of bounded Borel measurable functions, respectively, products of banach function spaces. These factorizations allow us to obtain integral dominations and lattice geometric properties and to present integral representations for abstract classes of multilinear operators. Finally, bringing together these ideas, some applications are shown, regarding for example summability properties and representation of multilinear maps as orthogonally additive n-homogeneous polynomials.
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