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...

A necessary and sufficient condition is given for a rearrangement invariant function space to contain a complemented isomorphic copy of l1(l2).

In this paper we give a sufficient condition on the pair of weights $(w,v)$ for the boundedness of the Weyl fractional integral $I_{\alpha }^{+}$ from $L^p\left(v\right)$ into $L^p\left(w\right)$. Under some restrictions on $w$ and $v$, this condition is also necessary. Besides, it allows us to show that for any $p:1\le p<\infty$ there exist non-trivial weights $w$ such that $I_{\alpha }^{+}$ is bounded from $L^p\left(w\right)$ into itself, even in the case $\alpha >1$.

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.

In this note we study some properties concerning certain copies of the classic Banach space $c_0$ in the Banach space $ℒ\left(X,Y\right)$ of all bounded linear operators between a normed space $X$ and a Banach space $Y$ equipped with the norm of the uniform convergence of operators.

Il lavoro presenta diverse caratterizzazioni degli spazi Lorentz-Zygmund generalizzati (GLZ) $L_{p,q;\alpha }\left(R\right)$, con $p,q\in \left(0,+\mathrm{\infty }\right]$, $m\in \mathbb{N}$, $\alpha \in {\mathbb{R}}^m$ e $\left(R,\mu \right)$ spazio misurato con misura $\mu \left(R\right)$ finita. Dato uno spazio misurato $\left(R,\mu \right)$ e $\alpha \in {\mathcal{R}}_{-}^m$ , otteniamo representazioni equivalenti per la (quasi-) norma dello spazio GLZ $L_{\mathrm{\infty },\mathrm{\infty };\alpha }\left(R\right)$. Inoltre, se $\left(R,\mu \right)$ è uno spazio misurato con misura finita e $\alpha \in {\mathcal{R}}_{+}^m$, viene presentata in termini di decomposizioni una norma equivalente per lo spazio $L_{1,1;\alpha }\left(R\right)$. Si dimostra che le norme equivalenti considerate per $L_{\mathrm{\infty },\mathrm{\infty };\alpha }\left(R\right)$, con $\left(R,\mu \right)$ uno spazio a misura finita, e la...

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.

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect.