Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where ${\nabla }_k$ is the gradient of order $k$, $ℳ$ is the Hardy-Littlewood maximal operator, and $I_z$ is the Riesz potential of order $z$, are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M(W_p^m\left({ℝ}^n\right)\to W_p^l\left({ℝ}^n\right))$ is described.

The paper discusses Problems 8 and 88 posed by Stanisław Mazur in the Scottish Book. It turns out that negative solutions to both problems are immediate consequences of the results of Peller [J. Operator Theory 7 (1982)]. We discuss here some quantitative aspects of Problems 8 and 88 and give answers to open problems discussed in a recent paper of Pełczyński and Sukochev in connection with Problem 88.

We prove sharp end forms of Holmstedt's reiteration theorem which are closely connected with a general form of Gehring's Lemma. Reverse type conditions for the Hardy-Littlewood-Pólya order are considered and the maximal elements are shown to satisfy generalized Gehring conditions. The methods we use are elementary and based on variants of reverse Hardy inequalities for monotone functions.

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.

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...

We investigate the multiplicative properties of the spaces $H^{\frac{n}{2},\frac{1}{2}}$ As in the case of the classical Sobolev spaces $H^{\frac{n}{2}}$ this space does not form an algebra. We investigate instead the space $H^{\frac{n}{2}}\cap L^{\infty }$ , more precisely a subspace of it formed by products of solutions of the homogeneous wave equation with data in $H^{\frac{n}{2}}$.

For $1\le p\le \infty$, precise conditions on the parameters are given under which the particular superposition operator $T:f\to \left|f\right|$ is a bounded map in the Besov space $B_{p,q}^s\left(R^1\right)$. The proofs rely on linear spline approximation theory.

The purpose of this paper is to give a characterization of the closure of the Lizorkin space in spaces of Beppo Levi type. As preparations for the proof, we establish the invariance of the Lizorkin space, and give local integral representations for smooth functions.