2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.We prove two results concerning the div-curl lemma without assuming any sort of exact cancellation, namely the divergence and curl need not be zero, and $$div\left(u^{-}{v}^{\to }\right)\in H^1\left(R^d\right)$$ which include as a particular case, the result of [3].

For a fusion Banach frame $(\{G_n,v_n\},S)$ for a Banach space $E$, if $(\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is a fusion Banach frame for $E^{*}$, then $(\{G_n,v_n\},S;\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is called a fusion bi-Banach frame for $E$. It is proved that if $E$ has an atomic decomposition, then $E$ also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.

We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${\psi (·-n)}_{n=-\infty }^{\infty }$, which is equivalent to the weight function ${\sum }_{n=-\infty }^{\infty }\left|\psi ̂(·+n)\right|²$ being > 0 a.e.

This paper is devoted to investigating the properties of multilinear $A_{P⃗}$ conditions and $A_{(P⃗,q)}$ conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of $A_{P⃗}$ and $A_{(P⃗,q)}$ classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear $A_{(P⃗,q)}$ classes in terms of the linear $A_p$ classes are established. These results essentially improve and extend...

A necessary and sufficient condition is given on the basis of a rare maximal function $M_l$ such that $M_{l}f\in L¹\left([0,1]\right)$ implies f ∈ L log L([0,1]).

Let $f\left(x\right)\sim \Sigma a_n{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_n\cos 2\pi nx$, where the $a{*}_n$ are the numbers $\left|a_n\right|$ rearranged so that $a_n^{*}\searrow 0$. Then for any convex increasing $\psi$, $\parallel \psi \left(\right|f{|}^2{\parallel }_1\le \parallel \psi \left(20\right|f*{|}^2{\parallel }_1$. The special case $\psi \left(t\right)=t^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel }_q\le 5\parallel f*{\parallel }_q$ an equivalent of Littlewood.