Some class of locally solid topologies (called uniformly $\mu$-continuous) on Köthe-Bochner spaces that are continuous with respect to some natural two-norm convergence are introduced and studied. A characterization of uniformly $\mu$-continuous topologies in terms of some family of pseudonorms is given. The finest uniformly $\mu$-continuous topology ${𝒯}_I^{\varphi }\left(X\right)$ on the Orlicz-Bochner space $L^{\varphi }\left(X\right)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).

Étude des propriétés des unions et intersections d’espaces $L^p\left(s\right)$ relatifs à un ensemble $S$ de mesures positives sur un groupe commutatif localement compact lorsque $S$ est invariant par translation ou stable par convolution.Dans des cas particuliers, on retrouve les propriétés d’espaces étudiés par A. Beurling et par B. Koremblium.On étudie aussi les espaces ${\ell }^p(L^{p^{\text{'}}})$ formés des fonctions appartenant localement à $L^{p^{\text{'}}}$ et qui ont un comportement ${\ell }^p$ à l’infini.

We study smoothness spaces generated by maximal functions related to the local approximation errors of integral operators. It turns out that in certain cases these smoothness classes coincide with the spaces $C_p^{\alpha }\left({ℝ}^d\right)$, 0 < p≤∞, introduced by DeVore and Sharpley [DS] by means of the so-called sharp maximal functions of Calderón and Scott. As an application we characterize the $C_p^{\alpha }\left({ℝ}^d\right)$ spaces in terms of the coefficients of wavelet decompositions.

We introduce variable exponent Fock spaces and study some of their basic properties such as boundedness of evaluation functionals, density of polynomials, boundedness of a Bergman-type projection and duality. We also prove that under the global log-Hölder condition, the variable exponent Fock spaces coincide with the classical ones.

The trace space of $W^{1,p\left(·\right)}\left(ℝⁿ\times [0,\infty )\right)$ consists of those functions on ℝⁿ that can be extended to functions of $W^{1,p\left(·\right)}\left(ℝⁿ\times [0,\infty )\right)$ (as in the fixed-exponent case). Under the assumption that p is globally log-Hölder continuous, we show that the trace space depends only on the values of p on the boundary. In our main result we show how to define an intrinsic norm for the trace space in terms of a sharp-type operator.

In this paper, we are going to characterize the space $\mathrm{BMO}\left({ℝ}^n\right)$ through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space $\mathrm{BMO}\left({ℝ}^n\right)$ by using various function spaces. For example, Ho obtained a characterization of $\mathrm{BMO}\left({ℝ}^n\right)$ with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known...

We give very short and transparent proofs of extrapolation theorems of Yano type in the framework of Lorentz spaces. The decomposition technique developed in Edmunds-Krbec (2000) enables us to obtain known and new results in a unified manner.

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum }_n{x}_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum }_k{x}_{n_k}$ (i.e. those with $n_{k+1}-n_k\to \infty$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

It is shown that an orthonormal wavelet basis for $L^2\left(ℝ\right)$ associated with a multiresolution is an unconditional basis for $L^p\left(ℝ\right)$, 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$, and denoted by $C_E^{\alpha }\left(x_0\right)$. We show how properties of $E$ are transferred into properties of $C_E^{\alpha }\left(x_0\right)$. Applications are given in multifractal analysis.