On multifractal formalism for self-similar measures with overlaps

Let mu be a self-similar measure generated by an IFS phi ={phi i}i=1 of similarities on Rd (d >= 1). When phi 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 is an element of[tau mu '(1),tau mu '(0+)]. We also verify the validity of the multifractal formalism of mu[tau mu '(infinity),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.