Assume $f_1,...,f_N$ a finite set of functions in $H^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I(f_1,...,f_N)$ generated by $f_1,...,f_N$, namely the property$$\left|f\right(z\left)\right|\le \alpha \left(\right|f_1\left(z\right)|+...+|f_N\left(z\right)\left|\right)\text{for}z\in D$$for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property$$\left|f\left(z\right)\right|\le C\underset{1\le j\le N}{max}\left|f_j\left(z\right)\right|\text{for}z\in D$$for some constant $C$, does not necessary imply that $f$ is...

Let D be the open unit disc and μ a positive bounded measure on [0,1]. Extending results of Mateljević/Pavlović and Shields/Williams we give Banach-space descriptions of the classes of all harmonic (holomorphic) functions f: D → ℂ satisfying ${ʃ}_0^1({ʃ}_0^{2\pi }|f\left(r{e}^{i\phi }\right){{|}^{p}d\phi )}^{q/p}d\mu \left(r\right)<\infty$.

One computes the joint and essential joint spectra of a pair of multiplication operators with bounded analytic functions on the Hardy spaces of the unit ball in ${ℂ}^n$.

Let A be a uniform algebra on X and σ a probability measure on X. We define the Hardy spaces $H^p\left(\sigma \right)$ and the $H^p\left(\sigma \right)$ interpolating sequences S in the p-spectrum ${ℳ}_p$ of σ. We prove, under some structural hypotheses on A and σ, that if S is a “dual bounded” Carleson sequence, then S is $H^s\left(\sigma \right)$-interpolating with a linear extension operator for s < p, provided that either p = ∞ or p ≤ 2. In the case of the unit ball of ℂⁿ we find, for instance, that if S is dual bounded in $H^{\infty }\left(\right)$ then S is $H^p\left(\right)$-interpolating with a linear...

In this article we study the weak type Hardy space of harmonic functions in the upper half plane ${\mathbf{R}}_{+}^{n+1}$ and we prove the $A$-integrability of singular integral transforms defined by Calderón-Zygmund kernels. This generalizes the corresponding result for Riesz transforms proved by Alexandrov.

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}{(}^2)$, $H^{(0,1)}{(}^2)$, or $H^{(1,1)}{(}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}{(}^2)$ or $H^{(0,1)}{(}^2)$ into weak-$L^1{(}^2)$, and also bounded from $H^{(1,1)}{(}^2)$ into $L^1{(}^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1}lo{g}^{+}L{(}^2)$, $L^1{\left(lo{g}^{+}L\right)}^2{(}^2)$, and $L^{\mu }{(}^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....