On the Fourier decay of multiplicative convolutions

We prove the following. Let ${\mu }_1,\dots ,{\mu }_n$ be Borel probability measures on $[-1,1]$ such that ${\mu }_j$ has finite $s_j$-energy for certain indices $s_j\in (0,1]$ with $s_1+\dots +s_n>1$. Then, the multiplicative convolution of the measures ${\mu }_1,\dots ,{\mu }_n$ has power Fourier decay: there exists a constant $\tau =\tau (s_1,\dots ,s_n)>0$ such that $$\mid \int e^{-2\pi i\xi \cdot x_1\cdots x_n}d{\mu }_1(x_1)\cdots d{\mu }_n(x_n)\mid \le \mathrm{\mid }\xi {\mathrm{\mid }}^{-\tau }$$ for sufficiently large $\mathrm{\mid }\xi \mathrm{\mid }$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_j$.