## Notes of Francois Vigneron’s lecture

Multifractal analysis on the Heisenberg group

Joint with Stéphane Seuret.

1. Motivation

1.1. Sources of multifractal analysis

Multifractal analysis is a toolbox for data analysis (textures, financial or experimental data, diophantine approimation,… see the program of our seminar at UPEC). It introduces classification parameters based on absence of regularity. Why the Heisenberg group ? People from image analysis ask about non isotropic textures.

1.2. Historic examples

Weierstrass’ example of a function ${W_h}$ which is ${C^h}$-Hölder but nowhere differentiable (for every ${0.

Riemann function which is differentiable at infinitely many rational points (but not all of them) and nowhere else,

$\displaystyle \begin{array}{rcl} \sum_n n^{-2}\sin(2\pi n^2 x). \end{array}$

Its multifractal spectrum was not computed before 1996 (S. Jaffard).

Both examples are self similar. A picture of Weierstrass’ function suggests that irregularity is spread all over. A picture of Riemann’s function looks very different : spikes, points with different left and right derivatives, differentiability point, all over.

Definition 1 The pointwise regularity exponent ${h_f(x_0)}$ of ${f}$ at ${x_0}$ is the largest ${\alpha>0}$ such that ${f}$ is, up to a polynomial, ${O(|x-x_0|)^\alpha)}$.

The multifractal spectrum is the function

$\displaystyle \begin{array}{rcl} d_f(h)=\mathrm{dim}_{\mathrm{Hausdorff}}(\{x\,;\, h_f (x)=h\}). \end{array}$

Example 1 Weierstrass function ${W_h}$ is monofractal : every point has pointwise regularity exponent ${h}$.

The multifractal spectrum of Riemann’s function is made of a segment joining ${h=1/2}$ and ${h=3/2}$ plus a point at ${h=3/2}$.

2. Heisenberg group

The strong anisotropy turns out not the make a big change for the specific questions we solve the existing techniques adapt rather easily. This is why we can state theorems

2.1. Wavelets

One can construct smooth, exponentially decaying functions ${\psi_{j,k}^{\epsilon}}$ concentrated at ${2^{-j}\circ k}$, where ${k\in {\mathbb Z}^3}$ (which is a subgroup), with vanishing moments, which form a basis of ${L^2}$.

2.2. Results

Theorem 2 (Global Hölder regularity) Let ${s=k+\sigma}$. A function belongs to ${C^s}$ iff its ${k}$-th horizontal derivatives are ${C^\sigma}$. Also iff its wavelet coefficients

Theorem 3 (Pointwise Hölder regularity) Let ${s=k+\sigma}$. If a function ${f}$ is ${C^s}$ at ${x_0}$, then

$\displaystyle \begin{array}{rcl} 2^{js}|d_{j,k}^\epsilon(f)|\leq C(1+2^j d(x_{j,k},x_0))^s . \end{array}$

Conversely, if this holds, then ${f}$ is ${C^t}$ at ${x_0}$ for all ${t.

2.3. Generic spectrum in H\” older and Besov classes

Theorem 4 Monofractal functions (at ${s}$) form a dense ${G_\delta}$ subset of ${C^s}$.

Definition 5 ${f\in B_{p,q}^s}$ if

$\displaystyle \begin{array}{rcl} \|2^{j(s-Q/p)}|d_{j,k}^\epsilon(f)|\|_{\ell^p(k)}\in \ell^{q}(j) . \end{array}$

Note that ${B_{p,q}^s \subset C^{s-Q/p}}$ for ${s>Q/p}$ and ${s\notin Q/p +{\mathbb N}}$. Here, ${Q=4}$.

Theorem 6 For a dense ${G_\delta}$ subset of ${B_{p,q}^s}$, the spectrum is a segment between ${(s-Q/p,0)}$ and ${(s,Q)}$. For all functions of ${B_{p,q}^s}$, the spectrum is below this segment.

Indeed, for the standard example of a Besov function (expressed in wavelet expansion), the pointwise Hölder exponent at ${x_0}$ is related to the dyadic approximation rate of ${x_0}$. Dimensions of isoapproximable sets can be computed. It is a special case of a very general result by Beresnevich, Dickinson and Velani (2006).