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...

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho$-integral, introduced by Jarník and Kurzweil. Let $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ be the space of all strongly $\rho$-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel ·\parallel$. We show that each element in the dual space of $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ can be represented as a strong $\rho$-integral. Consequently, we prove that $fg$ is strongly $\rho$-integrable on $E$ for each strongly $\rho$-integrable function $f$ if and only if $g$ is almost everywhere...

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions $c_p\left(X\right)$ onto $c_p\left(X\right)$ × ℝ$.Inparticular,$cp(X)$isnotlinearlyhomeomorphicto$cp(X)$\times ℝ$. One of these examples is compact. This answers some questions of Arkhangel’skiĭ.