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Inthesecondpartofthethesis,toobtainsymmetrizationinequalitiesonprobabilitymetricspacesthatadmitaconvexisoperimetricestimatorwhichincorporateintheirformulationtheisoperimetricestimatorandthatcanbeappliedtoprovideaunifiedtreatmentofsharpSobolev-Poincar´eandN

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Academic year: 2022

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Sobolev Inequalities: Isoperimetry and Symmetrization Walter Andr´es Ortiz Vargas

The first part of the thesis is devoted to obtain a Sobolev type embedding result for Besov spaces defined on a doubling metric space. This will be done by obtaining pointwise estimates between the special differencefµ∗∗(t)−fµ∗(t) (called oscillation of fµ∗) and theX−modulus of smoothness defined by

EX(f, r) ∶= ∥−∫B(x,r)∣f(x) −f(y)∣dµ(y)∥

X

.

(here fµ∗ is the decreasing rearrangement of f, fµ∗∗(t) = 1t∫0t fµ∗(s)ds, for all t>0 and X a rearrangement invariant space on Ω.

In the second part of the thesis, to obtain symmetrization inequalities on probability metric spaces that admit a convex isoperimetric estimator which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincar´e and Nash type inequalities.

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