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Weighted L Φ integral inequalities for operators of Hardy type

Steven BloomRon Kerman — 1994

Studia Mathematica

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for Φ 2 - 1 ( ʃ Φ 2 ( w ( x ) | T f ( x ) | ) t ( x ) d x ) Φ 1 - 1 ( ʃ Φ 1 ( C u ( x ) | f ( x ) | ) v ( x ) d x ) to hold when Φ 1 and Φ 2 are N-functions with Φ 2 Φ 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.

Optimal Sobolev imbedding spaces

Ron KermanLuboš Pick — 2009

Studia Mathematica

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.

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron KermanLuboš Pick — 2011

Studia Mathematica

We study imbeddings of the Sobolev space W m , ϱ ( Ω ) : = u: Ω → ℝ with ϱ ( α u / x α ) < ∞ 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, τ ϱ and σ ϱ , that are optimal with respect to the inclusions W m , ϱ ( Ω ) W m , τ ϱ ( Ω ) L σ ϱ ( Ω ) . General formulas for τ ϱ and σ ϱ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Ron KermanLuboš Pick — 2008

Studia Mathematica

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

A new algorithm for approximating the least concave majorant

Martin FrancůRon KermanGord Sinnamon — 2017

Czechoslovak Mathematical Journal

The least concave majorant, F ^ , of a continuous function F on a closed interval, I , is defined by F ^ ( x ) = inf { G ( x ) : G F , G concave } , x 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 𝒞 4 ( I ) , it can be well-approximated on I by a clamped cubic spline S . We show that S ^ is then a good approximation to F ^ . We give two examples, one to illustrate, the other to apply our algorithm.

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