In this article we study bilinear operators given by inner products of finite vectors of Calderón-Zygmund operators. We find that necessary and sufficient condition for these operators to map products of Hardy spaces into Hardy spaces is to have a certain number of moments vanishing and under this assumption we prove a Hölder-type inequality in the Hp space context.

We continue the study of multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We determine the set of all r ≤ 1 for which these operators map products of Lebesgue spaces Lp(Rn) into the Hardy spaces Hr(Rn). At the endpoint case r = n/(n + m + 1), where m is the highest vanishing moment of the multilinear operator, we prove a weak type result.

Let {Tt}t>0 be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space HA1 by means of a maximal function associated with the semigroup {Tt}t>0. Atomic and Riesz transforms characterizations of HA1 are shown.

Let ${K_t}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H{¹}_L$ if $\left|\right|su{p}_{t>0}|K_t{f\left(x\right)\left|\right||}_{L¹\left(dx\right)}<\infty$. We state conditions on V and $K_t$ which allow us to give an atomic characterization of the space $H{¹}_L$.

We present counterexamples to a conjecture of Böttcher and Silbermann on the asymptotic multiplicity of the Poisson kernel of the space $L^{\infty }\left(\partial D\right)$ and discuss conditions under which the Poisson kernel is asymptotically multiplicative.

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 mutual singularity problem for measures with restrictions on the spectrum is studied. The $d$-pluriharmonic Riesz product construction on the complex sphere is introduced. Singular pluriharmonic measures supported by sets of maximal Hausdorff dimension are obtained.

Bade, Curtis and Dales have introduced the idea of weak amenability. A commutative Banach algebra A is weakly amenable if there are no non-zero continuous derivations from A to A*. We extend this by defining an alternating n-derivation to be an alternating n-linear map from A to A* which is a derivation in each of its variables. Then we say that A is n-dimensionally weakly amenable if there are no non-zero continuous alternating n-derivations on A. Alternating n-derivations are the same as alternating...