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The notion of the Orlicz space is generalized to spaces of Banach-space valued functions. A well-known generalization is based on $N$ -functions of a real variable. We consider a more general setting based on spaces generated by convex functions defined on a Banach space. We investigate structural properties of these spaces, such as the role of the delta-growth conditions, separability, the closure of $ℒ^{\infty}$ , and representations of the dual space.

A pair of linear functional inequalities and a characterization of $L^{p}$ -norm

Dorota Krassowska, Janusz Matkowski (2005)

Annales Polonici Mathematici

It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of $L^{p}$ -norm is given.

A Positive Compact Operator.

D.H. Fremlin (1975)

Manuscripta mathematica

A potential operator and some ergodic properties of a positive $L_{\infty}$ contraction

K. A. Astbury (1976)

Annales de l'I.H.P. Probabilités et statistiques

A priori inequalities in $L^{\infty} (Ω)$ for solutions of elliptic equations in unbounded domains

Maurizio Chicco, Marina Venturino (1999)

Rendiconti del Seminario Matematico della Università di Padova

$A_{r}$ -condition for two weight functions and compact imbeddings of weighted Sobolev spaces

Bohumír Opic, Petr Gurka (1988)

Czechoslovak Mathematical Journal

A remark on complex powers of analytic functions

Giuseppe Zampieri (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Sia $K\subset\subset\mathbb{R}^{n}$ un compatto, $f\geq 0$ una funzione analitica all'intorno di $K$ , ed $m$ la massima molteplicità in $K$ degli zeri di $f$ ; si prova che la potenza $f^{\lambda}$ ( $\lambda\in\mathbb{C}$ , $Re\lambda>—\frac{1}{m}$ ) è integrabile in $K$ . L'estensione meromorfa dell'applicazione $\lambda\rightarrow f^{\lambda}$ da $Re\lambda>0$ a tutto $\mathbb{C}$ (con valori in $\mathcal{D}^{\prime}(K)$ anziché in $L^{1}(K)$ ) era già stata provata in [1] e [2].

A remark on E. Green’s paper “Completeness of $L^{p}$ -spaces over finitely additive set functions”

K. P. S. Bhaskara Rao, Vincenzo Aversa (1987)

Colloquium Mathematicae

A remark on finite-dimensional $P_{λ}$ -spaces

J. Bourgain (1982)

Studia Mathematica

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A convolution operator, bounded on $L^{q}(\mathbb{R}^{n})$ , is bounded on $L^{p}(\mathbb{R}^{n})$ , with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace $\mathbb{R}^{n}$ with a general non-commutative locally compact group $G$ . In this paper we give a simple construction of a convolution operator on a suitable compact group $G$ , wich is bounded on $L^{q}(G)$ for every $q\in[2,\infty)$ and is unbounded on $L^{p}(G)$ if $p\in[1,2)$ .

A remark on the multipliers of the Haar basis of L¹[0,1]

H. M. Wark (2015)

Studia Mathematica

A proof of a necessary and sufficient condition for a sequence to be a multiplier of the normalized Haar basis of L¹[0,1] is given. This proof depends only on the most elementary properties of this system and is an alternative proof to that recently found by Semenov & Uksusov (2012). Additionally, representations are given, which use stochastic processes, of this multiplier norm and of related multiplier norms.

A remark over the converse of Hölder inequality.

Cuenya, Héctor H., Levis, Fabián E. (1999)

Journal of Inequalities and Applications [electronic only]

A representation theorem for time-scale functionals

Marshall J. Leitman (1972)

Commentationes Mathematicae Universitatis Carolinae

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