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Capacitary Orlicz spaces, Calderón products and interpolation

100%
Silvestre P.
Banach Center Publications
|
2014
|
tom 102
|
nr 1
215-228
EN
These notes are devoted to the analysis on a capacity space, with capacities as substitutes of measures of the Orlicz function spaces. The goal is to study some aspects of the classical theory of Orlicz spaces for these spaces including the classical theory of interpolation.
2
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Resistance Conditions and Applications

63%
Kinnunen J., Silvestre P.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
276-294
EN
This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.
3
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Interpolation of quasicontinuous functions

51%
Cerdà J., Martín J., Silvestre P.
Banach Center Publications
|
2011
|
tom 95
|
nr 1
281-286
EN
If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K(t,f;L^p(C),L^∞(C))$ to quasicontinuous functions f ∈ QC is equivalent to $K(t,f;L^p(C) ∩ QC, L^∞(C) ∩ QC)$. We apply this result to identify the interpolation space $(L^{p₀,q₀}(C) ∩ QC,L^{p₁,q₁}(C) ∩ QC)_{θ,q}$.
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