Let ${𝒜}_{\beta }$ be the Beurling algebra with weight $(1+|n|{)}^{\beta }$ on the unit circle $𝕋$ and, for a closed set $E\subseteq 𝕋$, let $J_{{𝒜}_{\beta }}\left(E\right)=\{f\in {𝒜}_{\beta }:f=0\text{on}\text{a}\text{neighbourhood}\text{of}E\}$. We prove that, for $\beta \>\frac{1}{2}$, there exists a closed set $E\subseteq 𝕋$ of measure zero such that the quotient algebra ${𝒜}_{\beta }/\overline{J_{{𝒜}_{\beta }}\left(E\right)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras ${\lambda }_{\gamma }$ and the algebra $𝒜𝒞$ of absolutely continuous functions on $𝕋$, we characterize the closed sets $E\subseteq 𝕋$ for which the restriction algebras ${\lambda }_{\gamma }\left(E\right)$ and $𝒜𝒞\left(E\right)$ are generated by their idempotents.

As is known, color images are represented as multiple, channels, i.e. integer-valued functions on a discrete rectangle, corresponding to pixels on the screen. Thus, image compression, can be reduced to investigating suitable properties of such, functions. Each channel is compressed independently. We are, representing each such function by means of multi-dimensional, Haar and diamond bases so that the functions can be remembered, by their basis coefficients without loss of information. For, each...

By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.

Mathematics Subject Classification: 26A33, 34A37.In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential inclusions involving the Caputo fractional derivative. Both cases of convex and nonconvex valued right-hand side are considered. The topological structure of the set of solutions is also considered.

We study connections between the Boyd indices in Orlicz spaces and the growth conditions frequently met in various applications, for instance, in the regularity theory of variational integrals with non-standard growth. We develop a truncation method for computation of the indices and we also give characterizations of them in terms of the growth exponents and of the Jensen means. Applications concern variational integrals and extrapolation of integral operators.

On a metric measure space (X,ϱ,μ), consider the weight functions $w_{\alpha }\left(x\right)=\varrho {(x,z₀)}^{-\alpha ₀}$ if ϱ(x,z₀) < 1, $w_{\alpha }\left(x\right)=\varrho {(x,z₀)}^{-\alpha ₁}$ if ϱ(x,z₀) ≥ 1, $w_{\beta }\left(x\right)=\varrho {(x,z₀)}^{-\beta ₀}$ if ϱ(x,z₀) < 1, $w_{\beta }\left(x\right)=\varrho {(x,z₀)}^{-\beta ₁}$ if ϱ(x,z₀) ≥ 1, where z₀ is a given point of X, and let ${\kappa }_a:X\times X\to ℝ₊$ be an operator kernel satisfying ${\kappa }_a(x,y)\le c\varrho {(x,y)}^{a-d}$ for all x,y ∈ X such that ϱ(x,y) < 1, ${\kappa }_a(x,y)\le c\varrho {(x,y)}^{a-D}$ for all x,y ∈ X such that ϱ(x,y)≥ 1, where 0 < a < min(d,D), and d and D are respectively the local and global volume growth rate of the space X. We determine conditions on a, α₀, α₁, β₀, β₁ ∈ ℝ for the Hardy-Littlewood-Sobolev operator...

Le but de cet article est d’étendre les résultats classiques (inégalité de Hardy-Littlewood-Sobolev, inégalité de Hedberg) sur l’intégrale fractionnaire à deux types différents d’espaces métriques mesurés : les espaces métriques mesurés à mesure doublante d’une part, les espaces métriques mesurés à croissance polynomiale du volume d’autre part. Les deux résultats principaux que nous obtenons sont les suivants :Etant donné $(X,\rho ,\mu )$ un espace métrique mesuré de type homogène, étant donnés $p,q,\alpha \in \mathbf{R}$ tels que $1\le p\&lt;1/\alpha$, $1/q=1/p-\alpha$,...