Abstract
The classical Hölder regularity is restricted to locally bounded functions and takes only positive values. The local Lp regularity covers unbounded functions and negative values. Nevertheless, it has the same apparent regularity in all directions. In the present work, we study a recent notion of directional local Lp regularity introduced by Jaffard. We provide its characterization by a supremum of a wide range oriented anisotropic Triebel wavelet coefficients and leaders. In addition, we deduce estimates on the Hausdorff dimension of the set of points where the directional local Lp regularity does not exceed a given value. The obtained results are illustrated by some examples of self-affine cascade functions.
Highlights
Multifractal analysis describes geometrically and statistically the distribution of pointwise regularities of irregular functions f on Rm. It was first introduced in the context of the statistical study of fully developed turbulence in the mid 80’s [1]
In the asymptotic of small scales, the values taken into account should be localized more and more sharply around this line
The 2rVm elongated, centered on x, considered in [24]. This ellipsoid corresponds to the ball cRemntdereefidnaetdxbwy iρtkh(0ρ)k radius r, = 0, and where ρk is the quasi-norm on for x ≠ 0, ρk(x) is the unique value of r for which ∑mn=1(xn/rVn )2 = 1
Summary
Multifractal analysis describes geometrically and statistically the distribution of pointwise regularities (or singularities) of irregular functions f on Rm. To take into account pointwise directional Lp behavior in direction e, it is natural to define the local Lp regularity αp,e(y) at a point y in direction e as the p−exponent at 0 of the one variable function fy,e : t → f(y + te), that is, αp,e (y) = sup {α; fy,e ∈ Tαp (0)}. In the asymptotic of small scales, the values taken into account should be localized more and more sharply around this line These considerations motivated the following definition of Jaffard [24]. The 2rVm elongated , centered on x, considered in [24] This ellipsoid corresponds to the ball cRemntdereefidnaetdxbwy iρtkh(0ρ)k radius r, = 0, and where ρk is the quasi-norm on for x ≠ 0, ρk(x) is the unique value of r for which ∑mn=1(xn/rVn )2 = 1.
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