Fractal Characteristics of Measures; An Approach Via Function Spaces

Abstract

Let μ be a positive Radon measure in Rn with compact support Γ. Let Qjm be cubes with side-length 2-j+1 originating from the canonical tiling of Rn where j\in N0 and m\in Zn. If λ \in R, 0 < p \le ∞, 0 < q \le ∞, then μλpq is the mixed lq–lp-quasi-norm of the sequence 2j λ μ (Qjm). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces Bspq (Rn) and Fspq (Rn). In Theorem 1 we deal with the question when μλpq is an equivalent quasi-norm in some of these spaces (μ-property). If |Γ| = 0, then Sμ consists of those points (t,s) in the t–s-diagram in Figure 1 for which μ belongs to Bsp ∞ (Rn) with pt = 1. Theorem 2 deals with the interrelation of Sμ and μλpq. Some applications to truncated Riesz potentials, Bessel potentials and Fourier transforms of μ are given.