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.

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by ${\left(\varphi ₘ\right)}_{m\in \mathbb{N}₀}$. The system ${\left(\varphi ₘ\right)}_{m\in \mathbb{N}₀}$ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski has also considered some questions concerning expansions of functions from C(ℤₚ,ℂₚ) with respect to ${\left(\varphi ₘ\right)}_{m\in \mathbb{N}₀}$. This paper is a remark to Rutkowski’s paper. We define another system ${\left(hₙ\right)}_{n\in \mathbb{N}₀}$ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system...

In this paper we characterize those bounded linear transformations $Tf$ carrying $L^1\left({ℝ}^1\right)$ into the space of bounded continuous functions on ${ℝ}^1$, for which the convolution identity $T(f*g)=Tf·Tg$ holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

We prove that for f ∈ L ln⁺L(ℝⁿ) with compact support, there is a g ∈ L ln⁺L(ℝⁿ) such that (a) g and f are equidistributed, (b) $M_S\left(g\right)\in L¹\left(E\right)$ for any measurable set E of finite measure.