Self-similar measures pushed to non-degenerate curves satisfy optimal L^p-flattening estimates for their Fourier transforms, implying quantitative dimension improvement under convolution.
On bounded energy of convolution of fractal measures
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abstract
For all $s\in[0,1]$ and $t\in(0,s]\cup [2-s,2)$, we find the supremum of numbers $\omega\in(0,2)$ such that $\text{I}_\omega(\mu\ast\sigma) \lesssim 1$, where $\mu$ is any Borel measure on $B(1)$ with $\text{I}_t(\mu)\leq 1$ and $\sigma$ is any $(s,1)$-Frostman measure on a $C^2$-graph with non-zero curvature. As an application, we use this to show the sharp $L^6$-decay of Fourier transform of $\sigma$ when $s\in [\frac{2}{3}, 1]$.
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$L^2$-Flattening of Self-similar Measures on Non-degenerate Curves
Self-similar measures pushed to non-degenerate curves satisfy optimal L^p-flattening estimates for their Fourier transforms, implying quantitative dimension improvement under convolution.