BESOV-SPACES AND THE MULTIFRACTAL HYPOTHESIS

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 conjectured that the velocity field has local Holder exponents in a range [h(min), k(max)], with exponents <h occurring on a set S(h) with a Fractal dimension D(h). Heuristic reasoning led them to an expression for the scaling exponent z(p) of pth order as the Legendre transform of the codimension d-D(h). We show here that a part of the multifractal hypothesis is correct under even weaker assumptions: namely, if the velocity field has L(p)-mean Holder index s, i.e., if it les in the Besov space B-p(s,infinity), then local Holder regularity is satisfied. If s<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 exponent z(p), and, more generally, the maximal Besov index of order p, as s(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.