Besov spaces and the multifractal hypothesis

Abstract

Parisi and Frisch proposed some time ago an explanation for “multiscaling” of turbulent velocity structure functions in terms of a “multifractal hypothesis,” i.e., they conjecture that the velocity field has local Holder exponents in a range [hmin,hmax], with exponents <h occurring on a setS(h) with a fractal dimensionD(h). Heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the Legendre transform of the codimensiond-D(h). We show here that a part of the multifractal hypothesis is correct under even weaker assumptions: namely, if the velocity field hasL p -mean Holder indexs, i.e., if it lies in the Besov spaceB p s,∞ , then local Holder regularity is satisfied. Ifs<d/p, then the hypothesis is true in a generalized sense of Holder space with negative exponents and we discuss the proper definition of local Holder classes of negative index. Finally, if a certain “box-counting dimension” exists, then the Legendre transform of its codimension gives the scaling exponentz p , and, more generally, the maximal Besov index of order,p, ass p =z p /p. Our method of proof is derived from a recent paper of S. Jaffard using compactly-supported, orthonormal wavelet bases and gives an extension of his results. We discuss implications of the theorems for ensemble-average scaling and fluid turbulence.