Define $h^{\infty }\left(E\right)$ as the subspace of $C^{\infty }(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-\infty }(E*)$ consisting of all harmonic E*-valued functions g such that ${(1-|x\left|\right)}^{m}f$ is bounded for some m>0. Then the dual $h^{\infty }(E*)$ is represented by $h^{-\infty }(E*)$ through $⟨f,g{⟩}_0=li{m}_{r\to 1}{ʃ}_B⟨f\left(rx\right),g\left(x\right)⟩dx$, $f\in h^{-\infty }(E*),g\in h^{\infty }\left(E\right)$. This extends the results of S. Bell in the scalar case.

We introduce a new “weak” BMO-regularity condition for couples (X,Y) of lattices of measurable functions on the circle (Definition 3, Section 9), describe it in terms of the lattice $X^{1/2}{\left(Y^{\text{'}}\right)}^{1/2}$, and prove that this condition still ensures “good” interpolation for the couple $(X_A,Y_A)$ of the Hardy-type spaces corresponding to X and Y (Theorem 1, Section 9). Also, we present a neat version of Pisier’s approach to interpolation of Hardy-type subspaces (Theorem 2, Section 13). These two main results of the paper are...

Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X\otimes {̂}_{\epsilon }Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach...

We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_p\left(\right)$ does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, $H_p$ does not admit a Schauder basis with constant one.

One proves the density of an ideal of analytic functions into the closure of analytic functions in a $L^p\left(\mu \right)$-space, under some geometric conditions on the support of the measure $\mu$ and the zero variety of the ideal.

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 ${ʃ}_0^1({ʃ}_0^{2\pi }|f\left(r{e}^{i\phi }\right){{|}^{p}d\phi )}^{q/p}d\mu \left(r\right)<\infty$.

We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces $2^{}\oplus [0,\alpha ]$, the topological sums of Cantor cubes $2^{}$, with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C\left(2^{}\oplus [0,\alpha ]\right)$ spaces with ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.

We study the spaces $H_{\mu }\left(\Omega \right)=f:\Omega \to ℂholomorphic:{\int }_0^R{\int }_0^{2\pi }\left|f\left(r{e}^{i\phi }\right)\right|d\phi d\mu \left(r\right)<\infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{\mu }\left(\Omega \right)$ is either isomorphic to l₁ or to ${\left(\sum \oplus Aₙ\right)}_{\left(1\right)}$. Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.