We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in ${ℝ}^3$, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to $p=2$. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.

The paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional cubes into Zygmund spaces. This can be expressed in terms of tractability envelopes.

The paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional cubes into Zygmund spaces.

We show that the transference method of Coifman and Weiss can be extended to Hardy and Sobolev spaces. As an application we obtain the de Leeuw restriction theorems for multipliers.

Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions...

Nous étudions d’abord la transformation de Fourier sur les espaces ${\ell }^p(L^{p^{\text{'}}})$ qui sont formés de fonctions appartenant localement à $L^{p^{\text{'}}}$ et se comportant à l’infini comme des éléments de ${\ell }^p$. Si $1\le p,p^{\text{'}}\le 2$, les transformées de Fourier des éléments de ${\ell }^p(L^{p^{\text{'}}})$ appartiennent à ${\ell }^{q^{\text{'}}}(L^q)$. Dans les autres cas, nous donnons quelques résultats partiels.Nous montrons ensuite que ${\ell }^2(L^1)$ est le plus grand espace vectoriel solide de fonctions mesurables sur lequel la transformation de Fourier puisse se définir par prolongement par continuité.

The idempotent multipliers on Sobolev spaces on the torus in the L¹ and uniform norms are characterized in terms of the coset ring of the dual group of the torus. This result is deduced from a more general theorem concerning certain translation invariant subspaces of vector-valued function spaces on tori.

We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1,q}$. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...

This paper develops some Littlewood-Paley theory for Hermite expansions. The main result is that certain analogues of Triebel-Lizorkin spaces are well-defined in the context of Hermite expansions.

In [HS] the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type were introduced. In this paper, the Triebel-Lizorkin spaces on spaces of homogeneous type are generalized to the case where $p_0<p\le 1\le q<\infty$, and a new atomic decomposition for these spaces is obtained. As a consequence, we give the Littlewood-Paley characterization of Hardy spaces on spaces of homogeneous type which were introduced by the maximal function characterization in [MS2].

Suppose that μ is a Radon measure on ${ℝ}^d$, which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces ${Ḟ}_{pq}^s\left(\mu \right)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality...

The probability measure functor P carries open continuous mappings $f:Xonto\to Y$ of compact metric spaces into Q-bundles provided Y is countable-dimensional and all fibers $f^{-1}\left(y\right)$ are infinite. This answers a question raised by V. Fedorchuk.