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Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolicrational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Juliaset. We prove that $J$ always has positive Fourier dimension. The case where$J$ is included in a circle follows from a recent work of Sahlsten and Stevens,see arXiv:2009.01703. In the case where $J$ is not included in a circle, weprove that a large family of probability measures supported on $J$ exhibitpolynomial Fourier decay: our result applies in particular to the measure ofmaximal entropy and to the conformal measure.

Julia sets of hyperbolic rational maps have positive Fourier dimension