For every countable ordinal α, we construct an $l_1$-predual $X_{\alpha }$ which is isometric to a subspace of $C\left({\omega }^{{\omega }^{{\omega }^{\alpha }+2}}\right)$ and isomorphic to a quotient of $C\left({\omega }^{\omega }\right)$. However, $X_{\alpha }$ is not isomorphic to a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$.

For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ${\omega }_m\left(t\right)/\omega ₙ\left(t\right)\to \infty$ as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...

We give an extension of the commutator theorems of Jawerth, Rochberg and Weiss [9] for the real method of interpolation. The results are motivated by recent work by Iwaniek and Sbordone [6] on generalized Hodge decompositions. The main estimates of these authors are based on a commutator theorem for a specific operator acting on Lp spaces and through the use of the complex method of interpolation. In this note we give an extension of the Iwaniek-Sbordone theorem to general real interpolation scales....

We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L:ℇ\left(K\right)\to C^{\infty }[0,1]$. At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

Some boundedness and VMO results are proved for a function f integrable on a cube $Q_0$, starting from an integral bound.

Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.

We prove that for each countably infinite, regular space X such that $C_p\left(X\right)$ is a $Z_{\sigma }$-space, the topology of $C_p\left(X\right)$ is determined by the class $F_0\left(C_p\left(X\right)\right)$ of spaces embeddable onto closed subsets of $C_p\left(X\right)$. We show that $C_p\left(X\right)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set ${\Omega }_{\alpha }$ for the multiplicative Borel class $M_{\alpha }$ if $F_0\left(C_p\left(X\right)\right)=M_{\alpha }$. For each ordinal α ≥ 2, we provide an example $X_{\alpha }$ such that $C_p\left(X_{\alpha }\right)$ is homeomorphic to ${\Omega }_{\alpha }$.