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 ${\Omega }_{\alpha }$.

Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $$H_{\varphi }\left(\mu \right){\tilde{\otimes }}_{l}Y$$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$\left(H_{\varphi }\left(\mu \right){\tilde{\otimes }}_{l}Y\right)*$$ and $$H_{\varphi }\left(\mu \right)*{\tilde{\otimes }}_{l}Y*$$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým...

On examples we show a difference between a continuous and absolutely continuous norm in Banach function spaces.

$p$-convexity and $q$-concavity of a Banach lattice $L$ are characterized by factorization of multiplication operators from $L_q$ into $L_p$ through $L$. This characterization is applied to calculate the concavity type of Lorentz spaces.

Using the Haar-Kármán principle, approximate solutions of the basic boundary value problems are proposed and studied, which consist of piecewise linear stress fields on composite triangles. The torsion problem is solved in an analogous manner. Some convergence results are proven.

For a $C^1$-function $f$ on the unit ball $𝔹\subset {ℂ}^n$ we define the Bloch norm by ${\parallel f\parallel }_{𝔅}=sup\parallel \tilde{d}f\parallel ,$ where $\tilde{d}f$ is the invariant derivative of $f,$ and then show that $${\parallel f\parallel }_{𝔅}=\underset{\genfrac{}{}{0pt}{}{z,w\in 𝔹}{z\ne w}}{sup}{(1-|z|}^2{)}^{1/2}{(1-|w|}^2{)}^{1/2}\frac{\left|f\right(z)-f(w\left)\right|}{|w-P_{w}z-s_w{Q}_{w}z|}.$$

Let Γ be a closed set in ${ℝ}^n$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_1>0$ and $c_2>0$ such that $c_1{r}^d\le \mu \left(B(x,r)\right)\le c_2{r}^d$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_p\left(\Gamma \right)$, 0 < p ≤ ∞, with respect to...