J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $\Omega \subset {ℝ}^d$ is a smooth bounded domain, 1 ≤ p < ∞ and $f\in W^{1,p}\left(\Omega \right)$, then $li{m}_{s\nearrow 1}(1-s){\int }_{\Omega }{\int }_{\Omega }{\left(\right|f\left(x\right)-f\left(y\right)|}^p{)/(\left|\right|x-y\left|\right|}^{d+sp})dxdy=K{\int }_{\Omega }{\left|\nabla f\left(x\right)\right|}^{p}dx$, where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_p^{s,p}\left(\Omega \right)$. The purpose of this paper is to obtain analogous asymptotic formulae for some...

A new criterion of asymptotic periodicity of Markov operators on $L^1$, established in [3], is extended to the class of Markov operators on signed measures.

Estudiamos algunas cuestiones estructurales acerca del espacio localmente convexo HV∞, que está formado por funciones analíticas en el disco unidad abierto. Construimos una descomposición atómica de este espacio, usando un retículo de puntos del disco unidad que es más denso que el usual. También demostramos que HV∞ no es nuclear.

As a natural extension of $L^p$ Sobolev spaces, we consider Hardy-Sobolev spaces and establish an atomic decomposition theorem, analogous to the atomic decomposition characterization of Hardy spaces. As an application, we deduce several embedding results for Hardy-Sobolev spaces.

We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when this is not true.

Convolutional representations of the commutant of the partial integration operators in the space of continuous functions in a rectangle are found. Necessary and sufficient conditions are obtained for two types of representing functions, to be the operators in the commutant continuous automorphisms. It is shown that these conditions are equivalent to the requirement that the considered representing functions be joint cyclic elements of the partial integration operators.

Let α be an isometric automorphism of the algebra ${}_p$ of bounded linear operators in ${}^p[0,1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].

We investigate a scale of $BM{O}_{\psi }$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BM{O}_{\psi }$-$L_{\infty }$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned...

We characterize Baire-like spaces Cc(X,E) of continuous functions defined on a locally compact and Hewitt space X into a locally convex space E endowed with the compact-open topology.

We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).

We provide a structure theorem for Carnot-Carathéodory balls defined by a family of Lipschitz continuous vector fields. From this result a proof of Poincaré inequality follows.

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.