We prove some weighted endpoint estimates for some multilinear operators related to certain singular integral operators on Herz and Herz type Hardy spaces.

In the last decade it has become clear that one of the central themes within Gabor analysis (with respect to general time-frequency lattices) is a duality theory for Gabor frames, including the Wexler-Raz biorthogonality condition, the Ron-Shen duality principle and the Janssen representation of a Gabor frame operator. All these results are closely connected with the so-called Fundamental Identity of Gabor Analysis, which we derive from an application of Poisson's summation formula for the symplectic...

Suppose $\mu$ is a finite positive rotation invariant Borel measure on the open unit disc $\Delta$, and that the unit circle lies in the closed support of $\mu$. For $0\<p\<\infty$ the Bergman space$A_{\mu }^p$ is the collection of functions in $L^p\left(\mu \right)$ holomorphic on $\Delta$. We show that whenever a Gaussian power series $f\left(z\right)=\Sigma {\zeta }_n{a}_n{z}^n$ almost surely lies in $A_{\mu }^p$ but not in ${\bigcup }_{q\>p}{A}_{\mu }^p$, then almost surely: a) the zero set $Z\left(f\right)$ of $f$ is not contained in any $A_{\mu }^q$ zero set ($q\>p$, and b) $Z(f+1)\cup Z(f-1)$ is not contained in any $A_{\mu }^q$ zero set.

The $\omega$-weighted Besov spaces of holomorphic functions on the unit ball $B^n$ in $C^n$ are introduced as follows. Given a function $\omega$ of regular variation and $0<p<\infty$, a function $f$ holomorphic in $B^n$ is said to belong to the Besov space $B_p\left(\omega \right)$ if $${\parallel f\parallel }_{B_p\left(\omega \right)}^p={\int }_{B^n}{(1-|z|}^2{)}^p{\left|Df\left(z\right)\right|}^p\frac{\omega (1-|z\left|\right)}{{(1-|z|}^2{)}^{n+1}}d\nu \left(z\right)<+\infty ,$$ where $d\nu \left(z\right)$ is the volume measure on $B^n$ and $D$ stands for the fractional derivative of $f$. The holomorphic Besov space is described in the terms of the corresponding $L_p\left(\omega \right)$ space. Some projection theorems and theorems on existence of the inversions of these projections are proved. Also,...