The John–Nirenberg inequality in ball Banach function spaces and application to characterization of BMO

Our goal is to obtain the John–Nirenberg inequality for ball Banach function spaces X, provided that the Hardy–Littlewood maximal operator M is bounded on the associate space $X'$ X ′ by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.

Spatial Metaphor and the Development of Cross-Domain Mappings in Early Childhood

Spatial language is often used metaphorically to describe other domains, including time (long sound) and pitch (high sound). How does experience with these metaphors shape the ability to associate space with other domains? Here, we tested 3- to 6-year-old English-speaking children and adults with a cross-domain matching task. We probed cross-domain relations that are expressed in English metaphors for time and pitch (length-time and height-pitch), as well asrelations that are unconventional in English but expressed in other languages (size-time and thickness-pitch). Participants were tested with a perceptual matching task, in which theymatched between spatial stimuli and sounds of different durations or pitches, and a linguistic matching task, in which they matched between a label denoting a spatial attribute, duration, or pitch, and a picture or sound representing another dimension. Contrary to previous claims thatexperience with linguistic metaphors is necessary for children to make cross-domain mappings, children performed above chance for both familiar and unfamiliar relations in both tasks, as did adults. Children’s performance was also better when a label was provided for one of thedimensions, but only when making length-time, size-time, and height-pitch mappings (not thickness-pitch mappings). These findings suggest that, although experience with metaphorical language is not necessary to make cross-domain mappings, labels can promote these mappings,both when they have familiar metaphorical uses (e.g., English ‘long’ denotes both length and duration), and when they describe dimensions that share a common ordinal reference frame (e.g., size and duration, but not thickness and pitch).

A new perspective on the Riesz potential

This paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only \mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains {I_{\alpha}L^{p,\kappa}} under the conditions {1<p<\infty}, {1\leq q<\infty}, q(\kappa/p-\alpha)\leq\kappa\leq n, {0<\alpha<\min\{n,1+\kappa/p\}}, but also {\mathfrak{L}^{q,\lambda}} exists as an associate space under the condition {-q<\lambda<n}, where {I_{\alpha}L^{p,\kappa}} and {\mathfrak{L}^{q,\lambda}} are the Morrey–Sobolev and Campanato spaces on {\mathbb{R}^{n}} respectively. On the other hand, a nonnegative Radon measure μ is completely characterized to produce a continuous map {I_{\alpha}:L_{p,1}\to L^{q}_{\mu}} under the condition {1<p<\min\{q,{n}/{\alpha}\}} or {1<q\leq p<\min\{{q(n-\alpha p)}/({n-\alpha q(q-1)^{-1}}),{n}/{\alpha}\}}, where {L_{p,1}} and {L^{q}_{\mu}} are the {(p,1)}-Lorentz and {(q,\mu)}-Lebesgue spaces on {\mathbb{R}^{n}} respectively.

Embeddings and Duality Theorems for Weak Classical Lorentz Spaces

We characterize the weight functions u, v, w on (0,∞) such thatwhereAs an application we present a new simple characterization of the associate space to the space Γ∞(v), determined by the normwhere

Compactness Properties of Carleman and Hille-Tamarkin Operators

In this paper we study integral operators with domain a Banach function space Lρ1 and range another Banach function space Lρ2 or the space L0 of all measurable functions. Recall that a linear operator T from Lρ1 into L0 is called an integral operator if there exists a μ × v-measurable function T(x, y) on X × Y such thatSuch an integral operator is called a Carleman integral operator if for almost every x ∊ X the functionis an element of the associate space L′ρ1, i.e.,