We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.

We prove asymptotic formulas for the behavior of Gelfand and Kolmogorov numbers of Sobolev embeddings between Besov and Triebel-Lizorkin spaces of radial distributions. Our method works also for Weyl numbers.

Let $H^{\infty }\left(\right)$ denote the usual commutative Banach algebra of bounded analytic functions on the open unit disc of the finite complex plane, under Hadamard product of power series. We construct a Boehmian space which includes the Banach algebra A where A is the commutative Banach algebra with unit containing $H^{\infty }\left(\right)$. The Gelfand transform theory is extended to this setup along with the usual classical properties. The image is also a Boehmian space which includes the Banach algebra C(Δ) of continuous functions on...

We show that a free graded commutative Banach algebra over a (purely odd) Banach space $E$ is a Banach-Grassmann algebra in the sense of Jadczyk and Pilch if and only if $E$ is infinite-dimensional. Thus, a large amount of new examples of separable Banach-Grassmann algebras arise in addition to the only one example previously known due to A. Rogers.

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_d\ast g^{\ast d}+a_{d-1}\ast g^{\ast (d-1)}+\cdots +a₁\ast g+a₀=0$, where $a₀,...,a_d:\mathbb{N}\to ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form ${\sum }_{x\in X}f\left(x\right)e^{-sx}$ ($s\in {ℂ}^k$), where $X\subseteq {[0,\infty )}^k$ is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...

By a general Franklin system corresponding to a dense sequence of knots 𝓣 = (tₙ, n ≥ 0) in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots 𝓣, that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is a characterization of sequences 𝓣 for which the corresponding general Franklin system is a basis or an unconditional basis in H¹[0,1].

We show that each general Haar system is permutatively equivalent in $L^p\left([0,1]\right)$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p\left([0,1]\right)$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p\left({[0,1]}^d\right)$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases in $L^p\left({[0,1]}^d\right)$ for 1...

In this paper we give general methods of construction of various equivalent minimal pairs of compact convex sets that are not translates of one another.

By means of two general operations + and x, called pan-operations'', we build a new kind of integral. This formulation contains, as particular cases, both Choquet's and Sugeno's integrals.

Cet article est consacré à l’étude des espaces $A_{\omega }=ℱL^2({\mathbf{R}}^n;\omega \left(x\right)dx)$ qui sont des algèbres de Banach. On démontre que les multiplicateurs ponctuels de $A_{\omega }$ sont les fonctions qui appartiennent localement et uniformément à $A_{\omega }$ si et seulement si $A_{\omega }$ contient des fonctions à support compact.

We introduce the space $GBD$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $GBD$, which leads to a compactness result for the space $GSBD$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...