This paper is an informal presentation of material from [28]–[34]. The monotone envelopes of a function, including the level function, are introduced and their properties are studied. Applications to norm inequalities are given. The down space of a Banach function space is defined and connections are made between monotone envelopes and the norms of the down space and its dual. The connection is shown to be particularly close in the case of universally rearrangement invariant spaces. Next, two equivalent...

For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that $G{\left(U\right)}_i^{\text{'}}=(\mathscr{H}\left(U\right),{\tau }_{\delta })$. Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the ${\tau }_0$ and ${\tau }_{\omega }$ topologies on ℋ (U).

In this paper we give a characterization of uniform convergence on weakly compact sets, for sequences of homogeneous polynomials in terms of the Mosco convergence of their level sets. The result is partially extended for holomorphic functions. Finally we study the relationship with other convergences.

It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W(h_p,{\ell }_{\infty })$ to $W(L_p,{\ell }_{\infty })$. This implies the almost everywhere convergence of the Fejér means in a cone for all $f\in W(L₁,{\ell }_{\infty })$, which is larger than $L₁\left({ℝ}^d\right)$.

We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces $H^{p_k}$, $0<p_k\le 1$, to Lebesgue spaces $L^p$. These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral condition...

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval....

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.