Sets where lip is infinite Thomas Z urcher July 2017 lip = 1 / - - PowerPoint PPT Presentation

Sets where lip is infinite

Thomas Z¨ urcher July 2017

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Two story lines

◮ The story of lip.

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Two story lines

◮ The story of lip. ◮ Characterizing sets of non-differentiability.

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Rademacher and Stepanov

Theorem (Rademacher)

A Lipschitz function f : Rn → R is differentiable almost everywhere.

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Rademacher and Stepanov

Theorem (Rademacher)

A Lipschitz function f : Rn → R is differentiable almost everywhere.

Theorem (Stepanov)

If f : Rn → R satisfies Lip f (x) := lim sup

r→0

sup

y∈B(x,r)

|f (y) − f (x)| r = lim sup

y→x

|f (y) − f (x)| x − y < ∞ almost everywhere, then f is almost everywhere differentiable.

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Cheeger

Cheeger proves an analogue to Rademacher’s theorem in (quite general) metric measure spaces. This includes giving sense to the notion of differentiability almost everywhere in metric measure spaces.

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Keith

Keith gives a weaker condition for Cheeger’s result to hold. It involves the Lip − lip-inequality: Lip f (x) ≤ K lip f (x) := K lim inf

r→0

sup

y∈B(x,r)

|f (y) − f (x)| r for all Lipschitz functions f : X → R and almost all points x.

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Schioppa

In the correct, quite general setting: if a Lip − lip-inequality holds, then it holds for K = 1.

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Balogh–Rogovin–Z¨ urcher / Wildrick–Z¨ urcher

Stepanov’s theorem holds in metric measure spaces as well (where Rademacher holds). With Kevin Wildrick, we have a version for RNP-mappings as well (remember Sean Li’s talk).

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Balogh–Cs¨

• rnyei

In general, Stepanov’s theorem fails for lip.

Theorem (Balogh–Cs¨

• rnyei)

There exist a set A ⊂ [0, 1] of positive Lebesgue measure and a continuous function f : [0, 1] → R such that lip f (x) < ∞ for all x ∈ [0, 1], lip f ∈ Ls[0, 1] for all s < 1 and f is not differentiable at any point of A.

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Balogh–Cs¨

• rnyei

In general, Stepanov’s theorem fails for lip.

Theorem (Balogh–Cs¨

• rnyei)

There exist a set A ⊂ [0, 1] of positive Lebesgue measure and a continuous function f : [0, 1] → R such that lip f (x) < ∞ for all x ∈ [0, 1], lip f ∈ Ls[0, 1] for all s < 1 and f is not differentiable at any point of A.

Theorem (Balogh–Cs¨

• rnyei)

There exists a nowhere differentiable, continuous function f : [0, 1] → R such that lip f (x) = 0 for L1-a.e. x ∈ [0, 1].

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Theorem (Balogh–Cs¨

• rnyei)

Let Ω ⊂ Rn be a domain and f : Ω → R a continuous function. Assume that lip f (x) < ∞ for x ∈ Ω \ E, where the exceptional set E has σ-finite (n − 1)-dimensional Hausdorff measure. Assume that lip f ∈ Lp

loc(Ω) for some 1 ≤ p ≤ ∞. Then f ∈ W 1,p loc (Ω). If,

in addition p > n, we have lip f (x) = Lip f (x) = ∇f (x) for Ln-a.e. x ∈ Ω.

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Balogh–Cs¨

• rnyei, Heuristics

Sobolev functions behave well on almost every line. Show that only very few lines hit the exceptional set in an essential portion. The function is Sobolev on almost all lines, hence on the whole space (taking the correct integrability into account). In metric spaces: modulus of curves.

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Lorentz space version

With Kevin Wildrick we have a Lorentz space-version for metric measure spaces. Question: What is the correct assumption on the Poincar´ e inequality? We cannot use openendedness (see Luk´ aˇ s Mal´ y’s talk).

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Zahorski

A set E ⊂ R is the set of non-differentiability for a Lipschitz function if and only if E is a Gδσ set of measure zero.

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A related question

Given a set E ⊂ R of measure zero, is there a Lipschitz function that is not differentiable at any point of E?

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A related question

Given a set E ⊂ R of measure zero, is there a Lipschitz function that is not differentiable at any point of E?

• Yes. Each set of measure zero is contained in a Gδ set of measure

zero.

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Higher dimensions are different

Theorem (Preiss–Speight)

Suppose n > 1. Then there exists a Lebesgue null set N ⊂ Rn containing a point of differentiability for every Lipschitz function f : Rn → Rn−1. Also results by Michael Dymond And Olga Maleva. Compare to the talk by Andrea Pinamonti and Gareth Speight.

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Question

Can we characterize the sets where lip is infinite? By the way, the corresponding sets for Lip are the Gδ sets. There are results concerning the sets where derivatives are infinite (for example by Jarn´ ık).

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Status

With Zolt´ an Buczolich, Bruce Hanson, and Martin Rmoutil: Fσ-case almost done. The difficulty is in the construction of the function:

◮ countable sets ◮ perfect, nowhere dense sets ◮ union of nowhere dense, perfect sets ◮ union of nowhere dense sets ◮ Fσ-case

With David Preiss and Martin Rmoutil: Strategy for Fσδ is there, we need to work out the details. Difficulty: steps have to be carried out “simultaneously”.

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Typical results

Theorem (Banach)

The subset of all function in C([0, 1], ·∞) that are such that in no point t = 1 both right-sided derived numbers are finite is of the second category.

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Typical results

Theorem

The typical function f ∈ C([0, 1]d) satisfies lip f (x) = 0 at points

• f a residual set of full measure in [0, 1]d.

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Bibliography

Zolt´ an Buczolich, Bruce Hanson, Martin Rmoutil, and Thomas Z¨ urcher. On sets where lip f is finite. Work in progress. David Preiss, Martin Rmoutil, and Thomas Z¨ urcher. On sets where lip f is finite. Work even more in progress.

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