On the Fourier decay of multiplicative convolutions

We prove the following. Let $\mu_{1},\ldots,\mu_{n}$ 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 $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative convolution of the measures $\mu_{1},\ldots,\mu_{n}$ has power Fourier decay: there exists a constant $\tau = \tau(s_{1},\ldots,s_{n}) > 0$ such that $$\left| \int e^{-2\pi i \xi \cdot x_{1}\cdots x_{n}} \, d\mu_{1}(x_{1}) \cdots \, d\mu_{n}(x_{n}) \right| \leq |\xi|^{-\tau}$$ for sufficiently large $|\xi|$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_{j}$.