Absolutely Continuous Convolutions and Projections of Fractal Measures

Abstract

We develop a general criterion for establishing absolute continuity of convolutions of fractal measures on the line, and more generally of prescribed line projections of fractal measures in the plane. The crucial new ingredient is exact scaling covariance of the \(L^1\)-norm of Littlewood-Paley pieces of the projected measure, under affine renormalization. We apply this criterion in three key settings. First, we show that the convolution of two measures on the line is absolutely continuous whenever their dimensions sum to more than one, provided one is a self-conformal measure whose defining IFS is not \(C^2\)-conjugate to linear, and the other is either self-conformal or Ahlfors--David regular. Second, we show that every line projection of a planar complex-analytic self-conformal measure of dimension greater than one is absolutely continuous, under natural nonlinearity and nondegeneracy assumptions. Finally, we show that every line projection of a planar self-affine measure is absolutely continuous, under natural irreducibility and proximality assumptions, whenever the correlation dimension of the measure and the Frostman dimension of its Furstenberg measure sum to more than two.