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Lower semicontinuity of multiple µ-quasiconvex integrals

Ilaria Fragalà (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Lower semicontinuity results are obtained for multiple integrals of the kind ∫ ℝ n f ( x , ∇ μ u ) d μ , where μ is a given positive measure on ℝ n , and the vector-valued function u belongs to the Sobolev space H μ 1 , p ( ℝ n , ℝ m ) associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower...

Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

Guy C. David (2015)

Analysis and Geometry in Metric Spaces

A theorem of Lusin states that every Borel function onRis equal almost everywhere to the derivative of a continuous function. This result was later generalized to Rn in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.

McShane equi-integrability and Vitali’s convergence theorem

Jaroslav Kurzweil, Štefan Schwabik (2004)

Mathematica Bohemica

The McShane integral of functions f I → ℝ defined on an m -dimensional interval I is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.

Measure-preserving quality within mappings.

Stephen Semmes (2000)

Revista Matemática Iberoamericana

In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on which the given mapping is bilipschitz, with uniform bounds for the bilipschitz constant and the size of the subset. This has applications for boundedness of singular integral operators and uniform rectifiability of sets, as in [6], [7], [11], [13]. Some special cases of David's results, concerning projections...

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