M2 AAG : TD de Techniques d'analyse harmonique

TD associés au cours de Guy David (les notes de cours sont disponibles sur sa page web) effectués en 2016,2017,2018 et 2020 (en anglais).

Enoncés des TD (english version 2020)

• TD1 : Covering Lemmas

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  • We infer Vitali's theorem from Vitali's covering lemma.
  • We prove a weak version of Sard's Lemma applying Vitali's covering lemma.
  
  
Eléments de correction manuscrite.


• TD2 : Interpolation, a theorem of M. Riesz

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  We consider measurable 2π-periodic functions with values in \( \mathbb{C} \), which we identify with measurable functions on \( \mathbb{T} = \mathbb{R} / 2\pi\mathbb{Z} \). Recall that for \( p \in [1,+\infty) \), such a function \( f : \mathbb{R} \rightarrow \mathbb{C} \in {\rm L}^p \: \: (i.e. \: {\rm L}^p(\mathbb{T})) \) if $$ \| f \|_p := \left( \frac{1}{2\pi} \int_0^{2\pi} |f(t)|^p \, dt \right)^\frac{1}{p} \mbox{ is finite.} $$ We then define its Fourier coefficients and partial Fourier sums: $$ c_k(f) = \frac{1}{2\pi} \int_0^{2\pi} f(t) e^{-ikt} \, dt \quad \mbox{and} \quad S_n (f) = \sum_{k=-n}^n c_k(f) e^{ik\square} $$ The aim of the present exercises is to prove the following theorem dur to M. Riesz (1927):
  
  Theorem. \( \displaystyle \mbox{Let } p \in (1,+\infty) \mbox{ and } f \in {\rm L}^p(\mathbb{T}), \mbox{ then } S_n f \mbox{ converges to } f \mbox{ in } {\rm L}^p \: . \)
  
Eléments de correction manuscrite.


• TD3 : Sobolev spaces

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  Eléments de correction manuscrite.
  
  
• TD4 : Hausdorff Measure and Cantor Sets

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  Let \( \mathcal{H}^d \) denote the d-dimensional Hausdorff measure.
  • We construct a Cantor set \( K \subset [0,1] \) satifying for some \( d \in (0,1) \), 0 < \( \mathcal{H}^d (K) \) < \( +\infty \) and $$ \forall x \in K, \, \theta_\ast^d (K,x) = \liminf_{r \to 0_+} \frac{\mathcal{H}^d(K \cap B(x,r))}{(2 r)^d} = 0 \: . $$
  • We then define two Cantor sets \( K, \: K^\prime \subset [0,1] \) satisfying 0 < \(\mathcal{H}^d (K) \) < \( +\infty \) and 0 < \( \mathcal{H}^d (K^\prime) \) < \( +\infty \) and yet
    
    \( \mathcal{H}^{2d} (K \times K^\prime) = +\infty \: . \)
  
  Eléments de correction manuscrite.
  
  
• TD5 : BMO

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  Eléments de correction manuscrite.
  
  
• TD6 : Rectifiability

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  We construct a Besicovitch set, that is a Borel set \(B \subset \mathbb{R}^2 \) such that of null Lebesgue mesure \(\lambda(B)=0\) and containing a unit segment (and even a line here) in every direction.
  
  Eléments de correction manuscrite.
  
  
• TD7 : Whitney curve and extension

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  We show the existence of a (non rectifiable) curve \(\Gamma \subset \mathbb{R}^2\) and a function \(f \in {\rm C}^1( \mathbb{R}^2)\) that is not constant along \(\Gamma\), but such that \(\Gamma\) lies in the set of critical points of \(f\).
  
  Eléments de correction manuscrite.
  
  
• TD8 : A Lusin Type Theorem.

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  We focus on a result of G. Alberti (A Lusin Type Theorem, 1991).
  Theorem. Let \( \Omega \subset \mathbb{R}^n \) be an open set of finite measure and let \( f \in {\rm L}^1(\Omega, \mathbb{R}^n) \). Then, there exist \( u \in {\rm BV}(\mathbb{R}^n) \) and \( g : \Omega \rightarrow \mathbb{R}^n \) Borel functions such that \( Du = f \mathcal{L}^n + g \mathcal{H}^{n-1} \quad \mbox{and} \int |g| \: d \mathcal{H}^{n-1} \leq C \| f \|_{L^1} \: , \) where \( C \) is a constant depending on \( n \) only.
  
  Eléments de correction manuscrite.