Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

We prove unconditionality of general Franklin systems in $L^p\left(X\right)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

We give characterizations of weighted Besov-Lipschitz and Triebel-Lizorkin spaces with $A_{\infty }$ weights via a smooth kernel which satisfies “minimal” moment and Tauberian conditions. The results are stated in terms of the mixed norm of a certain maximal function of a distribution in these weighted spaces.

Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C(X,{\tau }_{\delta })$ of continuous real-valued functions on the space $(X,{\tau }_{\delta })$, where ${\tau }_{\delta }$ is the smallest Tikhonov topology on X for which $\tau \subseteq {\tau }_{\delta }$ and $C(X,{\tau }_{\delta })$ is von Neumann regular. The compact and metric spaces for which $G\left(X\right)=C(X,{\tau }_{\delta })$ are characterized. Necessary, and different sufficient, conditions...

The purpose of this paper is to provide a new characterization of the Sobolev space $W^{1,1}\left(ℝⁿ\right)$. We also show a new proof of the characterization of the Sobolev space $W^{1,p}\left(ℝⁿ\right)$, 1 ≤ p < ∞, in terms of Poincaré inequalities.

In this paper we prove a new convexity property for L₁ that resembles uniform convexity. We then develop a general theory that leads from the convexity property through normal structure to a fixed point property, via a theorem of Kirk. Applying this theory to L₁, we get the following type of normal structure: any convex subset of L₁ of positive diameter that is compact for the topology of convergence locally in measure, must have a radius that is smaller than its diameter. Indeed, a stronger result...