Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi}_{2}^{-1}(\u0283{\Phi}_{2}\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi}_{1}^{-1}(\u0283{\Phi}_{1}\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ to hold when ${\Phi}_{1}$ and ${\Phi}_{2}$ are N-functions with ${\Phi}_{2}\circ {\Phi}_{1}^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

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.

We study imbeddings of the Sobolev space
${W}^{m,\varrho}\left(\Omega \right)$: = u: Ω → ℝ with $\varrho ({\partial}^{\alpha}u/\partial {x}^{\alpha})$ < ∞ when |α| ≤ m,
in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, ${\tau}_{\varrho}$ and ${\sigma}_{\varrho}$, that are optimal with respect to the inclusions
${W}^{m,\varrho}\left(\Omega \right)\subset {W}^{m,{\tau}_{\varrho}}\left(\Omega \right)\subset {L}_{{\sigma}_{\varrho}}\left(\Omega \right)$.
General formulas for ${\tau}_{\varrho}$ and ${\sigma}_{\varrho}$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space ${W}^{m,\varrho}\left(\Omega \right)$ be compactly imbedded into the rearrangement-invariant space ${L}_{\sigma}\left(\Omega \right)$, 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 the Sobolev imbedding with the compactness of a certain Hardy operator from ${L}_{\varrho}(0,|\Omega \left|\right)$ into ${L}_{\sigma}(0,|\Omega \left|\right)$. The results are illustrated with examples in which ϱ and σ are both Orlicz norms...

The least concave majorant, $\widehat{F}$, of a continuous function $F$ on a closed interval, $I$, is defined by $$\widehat{F}\left(x\right)=inf\{G\left(x\right):G\ge F,\phantom{\rule{4pt}{0ex}}G\phantom{\rule{4.0pt}{0ex}}\text{concave}\},\phantom{\rule{1.0em}{0ex}}x\in I.$$
We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F\in {\mathcal{C}}^{4}\left(I\right)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\widehat{S}$ is then a good approximation to $\widehat{F}$. We give two examples, one to illustrate, the other to apply our algorithm.

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