In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an $\alpha$-Lipschitz operator from a compact metric space into a Banach space $A$ is defined and characterized in a natural way in the sence that $F:K\to A$ is a $\alpha$-Lipschitz operator if and only if for each $\sigma \in X^{*}$ the mapping $\sigma \circ F$ is a $\alpha$-Lipschitz function. The Lipschitz operators algebras $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result that $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are isometrically...

We introduce the notions of pointwise modulus of squareness and local modulus of squareness of a normed space X. This answers a question of C. Benítez, K. Przesławski and D. Yost about the definition of a sensible localization of the modulus of squareness. Geometrical properties of the norm of X (Fréchet smoothness, Gâteaux smoothness, local uniform convexity or strict convexity) are characterized in terms of the behaviour of these moduli.

In the paper, a sufficient and necessary condition is given for the locally uniformly weak star rotundity of Orlicz spaces with Orlicz norms.

Si dimostra che il funzionale ${\int }_{\mathrm{\Omega }}f(u,Du)dx$ è semicontinuo inferiormente su $W_{loc}^{1,1}(\mathrm{\Omega })$, rispetto alla topologia indotta da $L_{loc}^1(\mathrm{\Omega })$, qualora l’integrando $f(s,p)$ sia una funzione non-negativa, misurabile in $s$, convessa in $p$, limitata nell’intorno dei punti del tipo $(s,0)$, e tale che la funzione $s\mapsto f(s,0)$ sia semicontinua inferiormente su $𝐑$.

The Lukacs property of the free Poisson distribution is studied. We prove that if free and are free Poisson distributed with suitable parameters, then + and ${(+)}^{-1/2}{(+)}^{-1/2}$ are free. As an auxiliary result we compute the joint cumulants of and ${}^{-1}$ for free Poisson distributed . We also study the Lukacs property of the free Gamma distribution.

Necessary and sufficient conditions for URWC points and LURWC property are given in Orlicz sequence space lM.

In this article, we investigate new topological descriptions for two well-known mappings ${𝒵}_A$ and ${\Im }_A$ defined on intermediate rings $A\left(X\right)$ of $C\left(X\right)$. Using this, coincidence of each two classes of $z$-ideals, ${𝒵}_A$-ideals and ${\Im }_A$-ideals of $A\left(X\right)$ is studied. Moreover, we answer five questions concerning the mapping ${\Im }_A$ raised in [J. Sack, S. Watson, $C$ and $C^{*}$ among intermediate rings, Topology Proc. 43 (2014), 69–82].

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}{(}^2)$, $H^{(0,1)}{(}^2)$, or $H^{(1,1)}{(}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}{(}^2)$ or $H^{(0,1)}{(}^2)$ into weak-$L^1{(}^2)$, and also bounded from $H^{(1,1)}{(}^2)$ into $L^1{(}^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1}lo{g}^{+}L{(}^2)$, $L^1{\left(lo{g}^{+}L\right)}^2{(}^2)$, and $L^{\mu }{(}^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

DiPerna and Majda generalized Young measures so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in $L^p$-spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry)...

Mathematics Subject Classification: 44A05, 46F12, 28A78We prove that Dirac’s (symmetrical) delta function and the Hausdorff dimension function build up a pair of reciprocal functions. Our reasoning is based on the theorem by Mellin. Applications of the reciprocity relation demonstrate the merit of this approach.