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Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ 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 $$\widehat{F}\left(x\right)=inf\{G\left(x\right):G\ge F,G\text{concave}\},x\in I.$$ 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 $F\in {𝒞}^4\left(I\right)$, 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.