The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g{*}_{\lambda }$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts ${\mu }_S^{\varrho }$ and ${\mu }_{\lambda }^{*,\varrho }$ to S and $g{*}_{\lambda }$. The definition of ${\mu }_S^{\varrho }\left(f\right)$ is as follows: ${\mu }_S^{\varrho }\left(f\right)\left(x\right)=({ʃ}_{|y-x|<t}|1/t^{\varrho }{ʃ}_{\left|z\right|\le t}\Omega \left(z\right)/{\left(\right|z|}^{n-\varrho }{\left)f(y-z)dz\right|}^2\left(dydt\right)/\left(t^{n+1}\right){)}^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and...

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results...

Let $X$ be a complex Banach space. Recall that $X$ admits aif there exists a sequence ${\left\{X_n\right\}}_{n=1}^{\infty }$ of finite-dimensional subspaces of $X,$ such that every $x\in X$ has a unique representation of the form $x={\sum }_{n=1}^{\infty }{x}_n,$ with $x_n\in X_n$ for every $n.$ The finite-dimensional Schauder decomposition is said to beif, for every $x\in X,$ the series $x={\sum }_{n=1}^{\infty }{x}_n,$ which represents $x,$ converges unconditionally, that is, ${\sum }_{n=1}^{\infty }{x}_{\pi \left(n\right)}$ converges for every permutation $\pi$ of the integers. For short, we say that $X$ admits an unconditional F.D.D.We show that if X admits an unconditional...

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.

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ${ℂ}^n$ of codimension 1, $vo{l}_{2n-2}\left(D^{n-1}\right)\le vo{l}_{2n-2}\left(H\cap D^n\right)\le 2vo{l}_{2n-2}\left(D^{n-1}\right)$. The lower bound is attained if and only if H is orthogonal to the versor $e_j$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_j+\sigma e_k$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ${ℂ}^n$ with ${ℝ}^{2n}$; by $vo{l}_k\left(·\right)$ we denote the usual k-dimensional volume in ${ℝ}^{2n}$. The result is a complex counterpart of Ball’s [B1]...

Let $p$ and $q$ be two different prime integers such that $p\equiv q\equiv 3(mod8)$ with $(p/q)=1$, and $l$ a positive odd square-free integer relatively prime to $p$ and $q$. In this paper we investigate the unit groups of number fields $𝕃=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-l})$.

Let $\left(z_{\nu }\right)$ be a sequence in the upper half plane. If $1\&lt;p\le \infty$ and if $$y_{\nu }^{1/p}f(z_{\nu })=a_{\nu },\nu =1,2,...(*)$$ has solution $f\left(z\right)$ in the class of Poisson integrals of $L^p$ functions for any sequence $(a_{\nu })\in {\ell }^p$, then we show that $\left(z_{\nu }\right)$ is an interpolating sequence for $H^{\infty }$. If $f(z_{\nu })=a_{\nu }$, $\nu =1,2,...$ has solution in the class of Poisson integrals of BMO functions whenever $(a_{\nu })\in {\ell }^{\infty }$, then $\left(z_{\nu }\right)$ is again an interpolating sequence for $H^{\infty }$. A somewhat more general theorem is also proved and a counterexample for the case $p\le 1$ is described.

The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${\left(Tf\right)}^{(}x)=\lambda \left(x\right)f̂\left(x\right)$ $\left(\lambda \in C\left({ℝ}^n\right)\right)$ is bounded on $H^p\left({ℝ}^n\right)$ if and only if the restriction ${\lambda \left(\epsilon m\right)}_{m\in \Lambda }$ is an $H^p\left(T^n\right)$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in ${ℝ}^n$.