The variable exponent Hardy inequality ${∥x^{\beta (x)-1}{\int }_0^x\mathrm{‍}f(t)dt∥}_{L^{p(.)}(0,l)}\le C{∥x^{\beta (x)}f∥}_{L^{p(.)}(0,l)}$, $f\ge 0$ is proved assuming that the exponents $p:(0,l)\to (1,\infty )$, $\beta :(0,l)\to ℝ$ not rapidly oscilate near origin and $1/p^{\prime }(0)-\beta >0$. The main result is a necessary and sufficient condition on $p$, $\beta$ generalizing known results on this inequality.

Composition operators $C_{\phi }$ from Bloch-type ${ℬ}_{\alpha }$ spaces to $F\left(p,q,s\right)$ classes, from $F\left(p,q,s\right)$ to ${ℬ}_{\alpha }$, and from $F\left(p_1,q_1,0\right)$ to $F\left(p_2,q_2,s_2\right)$ are considered. The criteria for these operators to be bounded or compact are given. Our study also includes the corresponding hyperbolic spaces.

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.

We found that the classical Calderón-Zygmund singular integral operators are bounded on both the classical Hardy spaces and the product Hardy spaces. The purpose of this paper is to extend this result to a more general class. More precisely, we introduce a class of singular integral operators including the classical Calderón-Zygmund singular integral operators and show that they are bounded on both the classical Hardy spaces and the product Hardy spaces.

It is shown that for every positive integer n there exists a subnormal weighted shift on a directed tree (with or without root) whose nth power is densely defined while its ($n+1$)th power is not. As a consequence, for every positive integer n there exists a nonsymmetric subnormal composition operator C in an L ²-space over a σ-finite measure space such that Cⁿ is densely defined and $C^{n+1}$ is not.

This paper studies typical Banach and complete seminormed spaces of locally summable functions and their continuous functionals. Such spaces were introduced long ago as a natural environment to study almost periodic functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitable $L^p$ means. The supremum of such means defines a norm (or a seminorm, in the case of the full Marcinkiewicz space) that makes the respective spaces complete. Part of this paper is a review of the topological vector space structure, inclusion relations, and convolution operators. Then we expand and improve the deep theory due to Lau of representation of continuous functional and extreme points of the unit balls, adapt these results to Stepanoff spaces, and present interesting examples of discontinuous functionals that depend only on asymptotic values.

We first prove characterizations of $p$-uniform convexity and $q$-uniform smoothness. We next give a formulation on absolute normalized norms on ${ℂ}^2$. Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is not $p$-uniformly convex.