Let ψ and φ be analytic functions on the open unit disk $𝔻$ with φ($𝔻$) ⊆ $𝔻$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $𝒟$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations...

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$, with data in $L^1$. We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1,n}$.

A family of transformations on the set of all probability measures on the real line is introduced, which makes it possible to define new examples of convolutions. The associated central limit theorems are studied, and examples of the limit measures, related to the classical, free and boolean convolutions, are shown.

Some new examples of K-monotone couples of the type (X,X(w)), where X is a symmetric space on [0,1] and w is a weight on [0,1], are presented. Based on the property of w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X,X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is $t^{1/p}$ for some p ∈ [1,∞], then $X=L_p$. At...

We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$-dimensional Euclidean space ${ℝ}^m$. It is a “natural” extension of the variational McShane integral (the strong McShane integral) from $m$-dimensional closed non-degenerate intervals to open and bounded subsets of ${ℝ}^m$. We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our...

We show that every subset of L¹[0,1] that contains the nontrivial intersection of an order interval and finitely many hyperplanes fails to have the fixed point property for nonexpansive mappings.

Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of ${\left({\mu }^{⊠N}\right)}^{⊞N}$ as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free...

We obtain Hardy type inequalities $${\int }_0^{\infty }M\left(\omega \left(r\right)\left|u\left(r\right)\right|\right)\rho \left(r\right)dr⩽C_1{\int }_0^{\infty }M\left(\left|u\left(r\right)\right|\right)\rho \left(r\right)dr+C_2{\int }_0^{\infty }M\left(\left|u^{\text{'}}\left(r\right)\right|\right)\rho \left(r\right)dr,$$ and their Orlicz-norm counterparts $${∥\omega u∥}_{L^M({ℝ}_{+},\rho )}⩽{\tilde{C}}_1{∥u∥}_{L^M({ℝ}_{+},\rho )}+{\tilde{C}}_2{∥u^{\text{'}}∥}_{L^M({ℝ}_{+},\rho )},$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.