Ron Kerman

Brock University · Department of Mathematics

Fix the indices α and β, 1 < α < β < ∞, and suppose e is an Orlicz gauge or Lorentz gamma norm on the real-valued functions on a set X which are measurable with respect to a σ-finite measure μ on it. Set M(γ, X) := -f : X → R with sup λμ(-x ε X : |f(x)| > λ})1/γ < ∞}, γ = α, β. In this paper we obtain, as a special case, simple criteria to guarante...

Fix $b\in (0,\infty)$ and $p\in (1,\infty)$. Let $\phi$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\r_{p,\phi}$, at the measurable function $f:\R+\to\R+$ by $\rph(f):=[\int_0^bf^{**}(t)^p\phi(t)dt]^{\frac1p}$, in which $f^{**}(t):=t^{-1}\int_0^tf^{*}(s)ds$, where $f^*(t):=\mu_f^{-1}(t)$, with $\mu_f(s):=|\{x\...

This paper continues our study of Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. In it we characterize when the norms considered are optimal. Explicit expressions are given for the optimal partners corresponding to a given domain or range norm.

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace em...

We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ρ and σ, in order that the Sobolev space Wm,ρ(Ω) be compactly imbedded into the rearrangement-invariant space Lσ (Ω), where Ω is a bounded domain in ℝn with Lipschitz boundary and 1 ≤ m ≤ n - 1. In particular, we establish the equivalence of the compactness of t...

The Brudny˘ ı-Krugljak duality theory for the K-method is elaborated for a class of parameters derived from rearrangement-invariant spaces. As examples, con-crete expressions are given for the norms dual to certain interpolation spaces between two rearrangement-invariant spaces. These interpolation spaces are formed by the K-method using parameters...

The Brudny˘ ı-Krugljak duality theory for the K-method is elaborated for a class of parameters derived from rearrangement-invariant spaces. As examples, con-crete expressions are given for the norms dual to certain interpolation spaces between two rearrangement-invariant spaces. These interpolation spaces are formed by the K-method using parameters...

The aim of this paper is to study Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. We establish the equivalence of a Sobolev imbedding to the boundedness of a certain weighted Hardy operator. This Hardy operator is then used to prove the existence of rearrangement-invariant norms that are optimal in the i...