The Hausdorff and packing multifractal spectra of a Borel probability measure \(\mu\) on \({\mathbb{R}}^n\) are, respectively, defined as \[ f(\alpha)=\text{ dim}_H X(\alpha), \quad F(\alpha)=\text{ dim}_P X(\alpha), \quad\alpha\geq 0, \] where \(X(\alpha)\) are the layers of \(\mu\) with local dimension \(\alpha\), i.e.: \[ X(\alpha)=\left\{x: \lim_{r\rightarrow 0}{\log\mu B(x,r)\over \log r}=\alpha\right\}. \] The multifractal formalism – after [T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. I. Shraiman, “Fractal measures and their singularities: the characterization of strange sets”, Physical Review A (3)33, No. 2,1141-1151 (1986; Zbl 1184.37028); erratum: ibid. 34, No. 2, 1601 (1986)] – is said to hold when the functions \(f(\alpha)\) and \(F(\alpha)\) equal the Legendre transforms \(b^{*}(\alpha)\) and \(B^*(\alpha)\) of the so-called \(L^q\)-spectra of \(\mu\), \(b(q)\) and \(B(q)\), which are dimensional indexes defined from the behavior of the \(q\)-th moments of the measure.
In a significant paper, L. Olsen [“A multifractal formalism”, Adv. Math. 116, 82-196 (1995; Zbl 0841.28012)] developed a geometric measure-theoretic formalism to define \(b(q)\) and \(B(q)\) as the dimensional thresholds obtained in the usual way from families of Hausdorff and packing multifractal outer measures \(H_\mu^{q,t}\) and \(P_\mu^{q,t}\). Olsen proved that the upper bounds \(f(\alpha)\leq b^*(\alpha)\) and \(F(\alpha)\leq B^*(\alpha)\) hold for \(\alpha\) in a suitable interval in general. He derived the multifractal formalism under the assumption of the existence of a Gibbs measure at each state \(q\) for \(\mu\), a condition which is satisfied for self-conformal constructions with slight overlapping, but not expected to hold in general.
In the paper under review, the authors establish that the multifractal formalism holds under the condition that \(H_\mu^{q,B(q)}(\text{supp }\mu)>0\) – which is strictly weaker than the Gibbsian hypothesis on \(\mu\). This improves the result by Olsen, since the authors further exhibit a large class of measures satisfying the multifractal formalism while not having a Gibbs measure at any state \(q>0\). They also prove that the sufficient condition above is very close to be necessary for the formalism to hold.
Using the described results, two other related questions on multifractality posed by S. J. Taylor in [J. Fourier Anal. Appl. Spec. Iss., 553-568 (1995; Zbl 0886.28010)] are solved in the paper, namely, the authors prove the existence of non-exact dimensional measures with nontrivial multifractal spectra, and the existence of measures with real analytic spectra \(B(q)\) and with nowhere valid multifractal formalism.