LAUSR

Phase retrieval is concerned with recovering a function f from the absolute value of its Fourier transform $$|{\widehat{f}}|$$ | f ^ | . We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $$\begin{aligned} \Vert f-g\Vert _{L^2({\mathbb {R}}^n)} \le 2\cdot \Vert |{\widehat{f}}| - |{\widehat{g}}| \Vert _{L^2({\mathbb {R}}^n)} + h_f\left( \Vert f-g\Vert ^{}_{L^p({\mathbb {R}}^n)}\right) + J({\widehat{f}}, {\widehat{g}}), \end{aligned}$$ ‖ f - g ‖ L 2 ( R n ) ≤ 2 · ‖ | f ^ | - | g ^ | ‖ L 2 ( R n ) + h f ‖ f - g ‖ L p ( R n ) + J ( f ^ , g ^ ) , where $$1 \le p < 2$$ 1 ≤ p < 2 , $$h_f$$ h f is an explicit nonlinear function depending on the smoothness of f and J is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of $$L^p$$ L p for $$1 \le p < 2$$ 1 ≤ p < 2 : in this region $$L^p$$ L p cannot be used to control $$L^2$$ L 2 , our stability estimate has the flavor of an inverse Hölder inequality. It seems conceivable that the estimate is optimal up to constants.