发表机构
University of Haifa at Oranim; The Pennsylvania State University; Johns Hopkins University（奥拉尼姆海法大学; 宾夕法尼亚州立大学; 约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于L¹范数尺度协变性的判据，证明三类分形测度在维数条件下卷积或直线投影的绝对连续性。
AI 中文摘要
我们发展了一个一般性判据，用于建立直线上分形测度的卷积的绝对连续性，更一般地，用于建立平面上分形测度的指定直线投影的绝对连续性。关键的新要素是投影测度的Littlewood-Paley分量的L¹范数在仿射重正规化下具有精确的尺度协变性。我们在三个关键情境中应用此判据。首先，我们证明：当两个直线上测度的维数之和大于1时，它们的卷积是绝对连续的，前提是其中一个为自共形测度，其定义IFS不与C²共轭于线性系统，而另一个为自共形测度或Ahlfors-David正则测度。其次，我们证明：在自然的非线性和非退化假设下，维数大于1的平面复解析自共形测度的每一条直线投影都是绝对连续的。最后，我们证明：在自然的不可约性和近性假设下，当测度的关联维数与其Furstenberg测度的Frostman维数之和大于2时，平面自仿射测度的每一条直线投影都是绝对连续的。
英文摘要
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.
