In the present work, we give a new multifractal formalism for which the classical multifractal formalism does not hold. We precisely introduce and study a multifractal formalism based on the Hewitt-Stromberg measures and that this formalism is completely parallel to Olsen's multifractal formalism which based on the Hausdorff and packing measures.The multifractal formalism (1.1) has been proved rigorously for random and non-random self-similar measures [1,16,42,43, 51], for self-conformal measures [26,27,28,29,37,52], for self-affine measures [6,7,8,9,23,24,36,45] and for Moran measures [61,62,63,64]. We note that the proofs of the multifractal formalism (1.1) in the above-mentioned references [1,10,12,13,14,36,37,42,43,45,52] are all based on the same key idea. The upper bound for f µ (α) is obtained by a standard covering argument (involving Besicovitch's Covering Theorem 2000 Mathematics Subject Classification. 28A78, 28A80.

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Given two probability measures µ and ν on R n. We define the upper and lower relative multifractal box-dimensions of the measure µ with respect to the measure ν and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.

In this paper, we introduce the notion of upper metric mean dimension with potential on any subset (not necessarily compact or invariant) via Carathéodory-Pesin structures. We discuss several possible versions of upper measure-theoretic mean dimensions with potential and find conditions to make these notions coincide. In particular, we present a corresponding variational principle and an inverse variational principle.