On multifractal formalism for self-similar measures with overlaps

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Let $$\mu $$ be a self-similar measure generated by an IFS $$\varPhi =\{\phi _i\}_{i=1}^\ell $$ of similarities on $${{\mathbb {R}}}^d$$ ( $$d\ge 1$$ ). When $$\varPhi $$ is dimensional regular (see Definition 1.1), we give an explicit formula for the $$L^q$$ -spectrum $$\tau _\mu (q)$$ of $$\mu $$ over [0, 1], and show that $$\tau _\mu $$ is differentiable over (0, 1] and the multifractal formalism holds for $$\mu $$ at any $$\alpha \in [\tau _\mu '(1),\tau _\mu '(0+)]$$ . We also verify the validity of the multifractal formalism of $$\mu $$ over $$[\tau _\mu '(\infty ),\tau _\mu '(0+)]$$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the $$L^q$$ -spectrum of self-similar measures.