It is proved that if a Kothe sequence space $X$ is monotone complete and has the weakly convergent sequence coefficient WCS$\left(X\right)>1$, then $X$ is order continuous. It is shown that a weakly sequentially complete Kothe sequence space $X$ is compactly locally uniformly rotund if and only if the norm in $X$ is equi-absolutely continuous. The dual of the product space ${\left({⨁}_{i=1}^{\infty }{X}_i\right)}_{\Phi }$ of a sequence of Banach spaces ${\left(X_i\right)}_{i=1}^{\infty }$, which is built by using an Orlicz function $\Phi$ satisfying the ${\Delta }_2$-condition, is computed isometrically (i.e. the exact...

Criteria for full k-rotundity (k ∈ ℕ, k ≥ 2) and uniform rotundity in every direction of Calderón-Lozanovskiĭ spaces are formulated. A characterization of $H_{\mu }$-points in these spaces is also given.

We will present relationships between the modular ρ* and the norm in the dual spaces $\left(L_{\Phi }\right)*$ in the case when a Musielak-Orlicz space $L_{\Phi }$ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space $\left(L{⁰}_{\Phi }\right)*$ will be presented.

In this paper the spaces of type Sobolev-Morrey-W p,a,г,τl(Q,G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G∋R n satisfies the flexible λ-horn condition.

In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression...

In this paper the notions of uniformly upper and uniformly lower $\ell$-estimates for Banach function spaces are introduced. Further, the pair $(X,Y)$ of Banach function spaces is characterized, where $X$ and $Y$ satisfy uniformly a lower $\ell$-estimate and uniformly an upper $\ell$-estimate, respectively. The integral operator from $X$ into $Y$ of the form $$Kf\left(x\right)=\varphi \left(x\right){\int }_0^{x}k(x,y)f\left(y\right)\psi \left(y\right)\mathrm{d}y$$ is studied, where $k$, $\varphi$, $\psi$ are prescribed functions under some local integrability conditions, the kernel $k$ is non-negative and is assumed to satisfy certain additional...

For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^q,l^p)}^{\alpha }$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{\alpha }$. We study the closure ${(L^q,l^p)}_{c,0}^{\alpha }$ in ${(L^q,l^p)}^{\alpha }$ of the space of test functions (infinitely differentiable and with compact support in ${ℝ}^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W¹\left({(L^q,l^p)}^{\alpha }\right)$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space $W^{1,\alpha }$) and obtain...