If $x$ is a bounded function on $\mathbf{Z}$, the multiplier with symbol $x$ (denoted by $M_x)$ is defined by $(M_{x}f)\widehat{}=x\widehat{f}$, $f\in L^2\left(\mathbf{T}\right)$. We give some conditions on $x$ ensuring the “interpolation inequality” $\parallel M_{x}f{\parallel }_{L^p}\le C\parallel f{\parallel }_{L^1}^{\alpha }\parallel M_{x}f{\parallel }_{L^q}^{1-\alpha }$ (here $1\<p\<q$ and $\alpha =\alpha (p,q,x)$ is between 0 and 1). In most cases considered $M_x$ fails to have stronger $L^1$-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E\subset \mathbf{Z}$ every positive sequence in ${\ell }^2\left(E\right)$ can be majorized by the sequence $\{$ $\left|\widehat{f}\left(n\right)\right|{\}}_{n\in E}$ for some continuous funtion $f$ with spectrum...

Let $N_{\alpha }$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{\beta }$ are Möbius invariant, but $N_{\alpha }$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{\varphi }f=f◦\varphi$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{\varphi }:N_{\alpha }\to B$, $B\to Q_{\beta }$, $N_{\alpha }\to Q_{\beta }$ which is bounded resp. compact.

Given a function $f\left(t\right)\ge 0$ on $ℝ$ with ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2\left)\right)dt\<\infty$ and $\left|f\right(t)-f(t^{\text{'}}\left)\right|\le l|t-t^{\text{'}}|$, a procedure is exhibited for obtaining on $ℂ$ a (finite) superharmonic majorant of the function $$F\left(z\right):\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\left|𝔍z\right|}{|z-t{|}^2}f\left(t\right)dt-Al\left|𝔍z\right|,$$ where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f\left(t\right)$, positive and bounded away from 0 on $ℝ$, is such that ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2)dt\<\infty$ and denote,...

Using a description of the topology of the spaces ${\mathbf{E}}^{\text{'}}\left(\Omega \right)$ ($\Omega$ open convex subset of $R^n$) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$, $T\in {\mathbf{E}}^{\text{'}}$. We give applications to a class of distributions $T$ satisfying $$\text{cv.}\text{sing.}\text{supp.}S*T=\text{cv.}\text{sing.}\text{supp.}S+\text{cv.}\text{sing.}\text{supp.}T$$ for all $S\in {\mathbf{E}}^{\text{'}}$.

Pour les fonctions $f\left(x\right)$ dont les coefficients de Fourier $c_n$ satisfont à $\Sigma |c_n{|}^2{\lambda }_n\<\infty$, la capacité est évaluée pour l’ensemble où la fonction maximale satisfait à $f^{*}\left(x\right)\ge \lambda$.

In this paper we continue the study of the Fourier transform on ${\mathbf{R}}^n$, $n\ge 2$, analyzing the “almost-orthogonality” of the different directions of the space with respect to the Fourier transform. We prove two theorems: the first is related to an angular Littlewood-Paley square function, and we obtain estimates in terms of powers of $log\left(N\right)$, where $N$ is the number of equal angles considered in ${\mathbf{R}}^2$. The second is an extension of the Hardy-Littlewood maximal function when one consider cylinders of ${\mathbf{R}}^n$, $n\ge 2$,...