New Exponents for Pointwise Singularity Classification

Abstract

We introduce new tools for pointwise singularity classification: We investigate the properties of the two-variable function which is defined at every point as the p-exponent of a fractional integral of order t; new exponents are derived which are not of regularity type but give a more precise description of the behavior of the function near a singularity. We revisit several classical examples (deterministic and random) of multifractal functions for which the additional information supplied by this classification is derived. Finally, a new example of multifractal function is studied, where these exponents prove pertinent.

Appendix

We will prove the following result.

Theorem 6.5

Let p > 0 and $0 \leq t \leq \frac{1} {p}$ with $\frac{1} {p} -\frac{1} {q} = t$ . Suppose f ∈ T _p ^α . Then f ^(−t) belongs to T _q ^α+t .

Proof

By hypothesis f satisfies

$$\displaystyle{ \sum \limits _{\lambda '\subset 3\lambda _{j}(x_{0})}\vert c_{\lambda '}\vert ^{p}2^{-j'} \leq C2^{-j(\alpha p+1)} }$$

(79)

We will use the following inequalities whose proof we leave to the reader.

Lemma 6.6

Let I a set of countable indices and q > 0. We have

$$\displaystyle{ \sum \limits _{l\in \mathbb{Z}}(2^{lq} - 2^{(l-1)q})\sharp (k: \vert a_{ k}\vert \geq 2^{l}) \leq \sum \limits _{ k\in I}\vert a_{k}\vert ^{q} \leq \sum \limits _{ l\in \mathbb{Z}}(2^{(l+1)q} - 2^{lq})\sharp (\{k: \vert a_{ k}\vert \geq 2^{l}\}) }$$

(80)

A fractional integration of order t amounts to a change of wavelet basis and a multiplication of the coefficients c _j, k by 2^−jt. We want to compute

$$\displaystyle{ D_{\lambda }^{(t),q} = \left (\sum \limits _{\lambda ' \subset 3\lambda _{j}(x_{0})}\vert c_{\lambda '}\vert ^{q}2^{-j'tq}2^{-j'}\right )^{1/q} }$$

(81)

and prove that $D_{\lambda }^{(t),q} \leq C2^{-j(\alpha +t+\frac{1} {q})} = C2^{-j(\alpha + \frac{1} {p})}$

Following Lemma 6.6, since $\frac{1} {p} = \frac{1} {q} + t$,

$$\displaystyle{ (D_{\lambda }^{(t),q})^{q} \leq \sum \limits _{ l\in \mathbb{Z}}(2^{(l+1)q} - 2^{lq})\sharp (\{\lambda ' \subset 3\varLambda _{ j}(x_{0}): \vert c_{\lambda '}\vert 2^{-j't}2^{-\frac{j'} {q} } \geq 2^{l}\}) }$$

(82)

$$\displaystyle{ \leq \sum \limits _{l\in \mathbb{Z}}(2^{(l+1)q} - 2^{lq})\sharp (\{\lambda ' \subset 3\varLambda _{ j}(x_{0}): \vert c_{\lambda '}\vert 2^{-\frac{j'} {p} } \geq 2^{l}\}) }$$

(83)

Remark that by (79) we have $\vert c_{\lambda '}\vert 2^{-\frac{j'} {p} } \leq C2^{-j(\alpha + \frac{1} {p})}.$ Thus if $2^{l} \leq \vert c_{\lambda '}\vert 2^{-\frac{j'} {p} }$ we have $2^{l} \leq C2^{-j(\alpha + \frac{1} {p})}$, which yields $l \leq -j(\alpha +\frac{1} {p}) + J_{0} \leq l_{1}$ with $J_{0} \in \mathbb{Z}$ a constant independent of j and j′. Following (79) we have

$$\displaystyle{ \sum \limits _{\lambda '\subset 3\lambda _{j}(x_{0})}\vert c_{\lambda '}\vert ^{p}2^{-j'} \leq C2^{-j(\alpha p+1)},\qquad \mbox{ so that} }$$

(84)

$$\displaystyle{ \sum \limits _{l=-\infty }^{\infty }(2^{lp} - 2^{(l-1)p})\sharp (\{\lambda ' \subset 3\varLambda _{ j}(x_{0}): \vert c_{\lambda '}\vert 2^{-\frac{j'} {p} } \geq 2^{l}\}) \leq C2^{-j(\alpha p+1)}. }$$

(85)

Thus for all $l \in \mathbb{Z}$,

$$\displaystyle\begin{array}{rcl} (1 - 2^{-p})2^{lp}\sharp (\{\lambda ' \subset 3\varLambda _{ j}(x_{0}): \vert c_{\lambda '}\vert 2^{-\frac{j'} {p} } \geq 2^{l}\})& \leq & C2^{-j(\alpha p+1)} {}\\ \sharp (\{\lambda ' \subset 3\varLambda _{j}(x_{0}): \vert c_{\lambda '}\vert 2^{-\frac{j'} {p} } \geq 2^{l}\})& \leq & \frac{C} {1 - 2^{-p}}2^{-lp}2^{-j(\alpha p+1)}. {}\\ \end{array}$$

Since q − p > 0, this yields,

$$\displaystyle\begin{array}{rcl} (D_{\lambda }^{(t),q})^{q}& \leq & \sum \limits _{ l=-\infty }^{l_{1} }(2^{(l+1)q} - 2^{lq})\sharp (\{\lambda ' \subset 3\varLambda _{ j}(x_{0}): \vert c_{\lambda '}\vert 2^{-\frac{j'} {p} } \geq 2^{l}\}) {}\\ & \leq & \frac{C(2^{q} - 1)} {1 - 2^{-p}} \sum \limits _{l=-\infty }^{l_{1} }2^{lq-lp}2^{-j(\alpha p+1)} \leq C'2^{-j(\alpha p+1)}2^{-j(q-p)(\alpha + \frac{1} {p})} {}\\ & \leq & C'2^{-j(\alpha p+1+\alpha q-p\alpha -1+\frac{q} {p})} = C'2^{-j(\alpha q+\frac{q} {p})}, {}\\ \end{array}$$

which yields the result.