Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_{\epsilon }\left(f₁\right)\cdots B_{\epsilon }\left(fₙ\right)$ for all ε > 0 ? ($B_{\epsilon }$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to...

In this brief note, we see that if $A$ is a proper uniform algebra on a compact Hausdorff space $X$, then $A$ is flat.

Nous montrons que $h^2(𝔻,L^1\left(𝕋\right))$ admet une norme équivalente $LUR,$ ce qui répond négativement à une question de Dowling, Hu et Smith. Puis nous obtenons une propriété de stabilité des opérateurs de Radon-Nikodym analytique. Motivés par l’identification entre $h^p(𝔻,X)$ et $V{B}^p(𝕋,X)$ où $X$ est un espace de Banach, pour un groupe abélien compact métrisable $G$, son dual $\Gamma$, et ${\Lambda }_2\subset {\Lambda }_1\subset \Gamma$, nous prouvons que, si l’espace $V{B}_{{\Lambda }_1}^p(G,X)/V{B}_{{\Lambda }_2}^p(G,X)$ a la propriété $Kadec-Klee-{\beta }^{\prime }-\omega$, alors il coincïde avec $L_{{\Lambda }_1}^p(G,X)/L_{{\Lambda }_2}^p(G,X),$$...$

Assuming ⋄, we construct a connected compact topological space K such that for every closed L ⊂ K the Banach space C(L) has few operators, in the sense that every operator on C(L) is multiplication by a continuous function plus a weakly compact operator. In particular, C(K) is indecomposable and has continuum many non-isomorphic indecomposable quotients, and K does not contain a homeomorphic copy of βℕ. Moreover, assuming CH we construct a connected compact K where C(K) has few...

We establish weighted Hardy-Littlewood inequalities for radial derivative and fractional radial derivatives on bounded symmetric domains.

2000 Mathematics Subject Classification: Primary 46E15, 54C55; Secondary 28B20.For weakly compact subsets of Hilbert spaces K, we study the existence of totally disconnected spaces L, such that C(K) is isomorphic to C(L). We prove that the space C(BH ) admits a Pełczyński decomposition and we provide a starshaped weakly compact K, subset of BH with non-empty interior in the norm topology, and such that C(K) ~= C(L) with L totally disconnected.Research partially supported by EPEAEK program “Pythagoras”....

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...