Measuring families of curves by approximation modulus Jan MAL´ Y a joint work with Olli Martio and Vendula Honzlov´ a Exnerov´ a Faculty of Mathematics and Physics, Charles University, Prague Geometric Measure Theory Mathematics Institute, University of Warwick July 10-14, 2017 Jan Mal´ y (Prague) Approximation modulus 1 / 17
Modulus Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Jan Mal´ y (Prague) Approximation modulus 2 / 17
Modulus Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Jan Mal´ y (Prague) Approximation modulus 2 / 17
Modulus Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Modulus of the family of all curves which reaches E from the boundary of Ω is the capacity of E in Ω (Ziemer 1969). Jan Mal´ y (Prague) Approximation modulus 2 / 17
Modulus Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Modulus of the family of all curves which reaches E from the boundary of Ω is the capacity of E in Ω (Ziemer 1969). Recent alternative: Probability measures on a space of curves (Ambrosio, Gigli and Savar´ e 2013, 2014). The two approaches are mutually dual (Ambrosio, Di Marino and Savar´ e 2015). Jan Mal´ y (Prague) Approximation modulus 2 / 17
Modulus Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Modulus of the family of all curves which reaches E from the boundary of Ω is the capacity of E in Ω (Ziemer 1969). Recent alternative: Probability measures on a space of curves (Ambrosio, Gigli and Savar´ e 2013, 2014). The two approaches are mutually dual (Ambrosio, Di Marino and Savar´ e 2015). A similar object: Alberti representation - invented for another purposes. Jan Mal´ y (Prague) Approximation modulus 2 / 17
Definition Let Γ be a family of nowhere constant rectifiable parametric curves into a metric measure space ( X , | · − · | , m ). We say that a function ρ : X → [0 , ∞ ] is an admissible function for Γ if � ρ ds ≥ 1 for each γ ∈ Γ . γ Let p ∈ [1 , ∞ ). We define the L p , q -modulus of Γ as � � � ρ � p M L p , q (Γ) = inf L p , q : ρ is admissible for Γ . We simplify the symbol M L p , p as M p . Jan Mal´ y (Prague) Approximation modulus 3 / 17
Definition Let Γ be a family of nowhere constant rectifiable parametric curves into a metric measure space ( X , | · − · | , m ). We say that a function ρ : X → [0 , ∞ ] is an admissible function for Γ if � ρ ds ≥ 1 for each γ ∈ Γ . γ Let p ∈ [1 , ∞ ). We define the L p , q -modulus of Γ as � � � ρ � p M L p , q (Γ) = inf L p , q : ρ is admissible for Γ . We simplify the symbol M L p , p as M p . Recall: � ∞ � 1 q ; q � α q − 1 m ( { f > α } ) p d α � f � L p , q = in particular � f � L p , p = � f � L p . p 0 Jan Mal´ y (Prague) Approximation modulus 3 / 17
Approximation modulus Definition (Martio 2016) Let Γ be a family of nowhere constant rectifiable parametric curves into a metric measure space ( X , | · − · | , m ). We say that a sequence of functions ( ρ j ) j , ρ j : X → [0 , ∞ ] is an admissible sequence for Γ if � lim inf ρ j ds ≥ 1 for each γ ∈ Γ . j γ Let p ∈ [1 , ∞ ). We define the L p , q -approximation modulus of Γ as � � � ρ j � p AM L p , q (Γ) = inf lim inf L p , q : ( ρ j ) j is admissible for Γ . j We simplify the symbols AM L p , p as AM p and AM 1 as AM . Jan Mal´ y (Prague) Approximation modulus 4 / 17
More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable observation is the following Jan Mal´ y (Prague) Approximation modulus 5 / 17
More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable observation is the following Theorem (HEMM) Let F be a reflexive Banach function space. Then the M F -modulus and the AM F -modulus are the same. Jan Mal´ y (Prague) Approximation modulus 5 / 17
More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable observation is the following Theorem (HEMM) Let F be a reflexive Banach function space. Then the M F -modulus and the AM F -modulus are the same. We focus our attention to Lorentz spaces, as the AM L p , 1 moduli fit to measuring families of curves related to ( n − p )-dimensional sets. Jan Mal´ y (Prague) Approximation modulus 5 / 17
More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable observation is the following Theorem (HEMM) Let F be a reflexive Banach function space. Then the M F -modulus and the AM F -modulus are the same. We focus our attention to Lorentz spaces, as the AM L p , 1 moduli fit to measuring families of curves related to ( n − p )-dimensional sets. Remark The AM -modulus has been originally introduced by Martio 2016 et 2016 as a modulus corresponding to BV spaces. Actually, he developed an alternative to Miranda BV-spaces on metric measure spaces. (Martio’s spaces are bigger; it is not investigated whether they are equivalent.) AM F -moduli for other function spaces were investigated later by Honzlov´ a Exnerov´ a, M. and Martio. Jan Mal´ y (Prague) Approximation modulus 5 / 17
Theorem (Martio) Let u ∈ BV ( R n ) (a precise representative). Then for AM 1 almost every Lipschitz curve γ , the composition u ◦ γ is of bounded variation. Jan Mal´ y (Prague) Approximation modulus 6 / 17
Theorem (Martio) Let u ∈ BV ( R n ) (a precise representative). Then for AM 1 almost every Lipschitz curve γ , the composition u ◦ γ is of bounded variation. The result above fails if we replace AM 1 by M 1 . Jan Mal´ y (Prague) Approximation modulus 6 / 17
Theorem (Martio) Let u ∈ BV ( R n ) (a precise representative). Then for AM 1 almost every Lipschitz curve γ , the composition u ◦ γ is of bounded variation. The result above fails if we replace AM 1 by M 1 . Example (HEMM) Let u be a characteristic function of the unit cube. Then the M 1 modulus of the family of Lipschitz curves Γ = { γ : u ◦ γ fails to be of bounded variation } is infinity. Jan Mal´ y (Prague) Approximation modulus 6 / 17
Estimates of moduli of concrete curve families In the subsequent results, the size of sets in consideration is measured by the lower Minkowski content, or by the Dirichlet-Lorentz capacity (like the Newtonian-Lorentz capacity, but without the lower order term). Jan Mal´ y (Prague) Approximation modulus 7 / 17
Estimates of moduli of concrete curve families In the subsequent results, the size of sets in consideration is measured by the lower Minkowski content, or by the Dirichlet-Lorentz capacity (like the Newtonian-Lorentz capacity, but without the lower order term). For simplicity, we can imagine that E is a nonempty part of a k -dimensional C 1 -surface in R n , where k ∈ { 0 , 1 , . . . , n − 1 } , p = n − k , and E is compact (for the upper estimate) and relatively open (for the lower estimate). Jan Mal´ y (Prague) Approximation modulus 7 / 17
We suppose that the measure m on X is doubling. Definition Let E ⊂ X . Define Γ( E ) = { γ : γ meets E } , Γ ∞ ( E ) = { γ : γ meets E infinitely times } , � � d ( γ ( t ) , E ) Γ τ ( E ) = γ : γ (0) ∈ E , lim t → 0+ = 0 (“tangential”). t Jan Mal´ y (Prague) Approximation modulus 8 / 17
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