On the existence of optimizers for time-frequency concentration problems

We consider the problem of the maximum concentration in a fixed measurable subset $\Omega\subset\mathbb{R}^{2d}$ of the time-frequency space for functions $f\in L^2(\mathbb{R}^{d})$. The notion of concentration can be made mathematically precise by considering the $L^p$-norm on $\Omega$ of some time-frequency distribution of $f$ such as the ambiguity function $A(f)$. We provide a positive answer to an open maximization problem, by showing that for every subset $\Omega\subset\mathbb{R}^{2d}$ of finite measure and every $1\leq p<\infty$, there exists an optimizer for $$\sup\{\|A(f)\|_{L^p(\Omega)}:\ f\in L^2(\mathbb{R}^{d}),\ \|f\|_{L^2}=1 \}.$$ The lack of weak upper semicontinuity and the invariance under time-frequency shifts make the problem challenging. The proof is based on concentration compactness with time-frequency shifts as dislocations, and certain integral bounds and asymptotic decoupling estimates for the ambiguity function. We also discuss the case $p=\infty$ and related optimization problems for the time correlation function, the cross-ambiguity function with a fixed window, and for functions in the modulation spaces $M^q(\mathbb{R}^{d})$, $0<q<2$, equipped with continuous or discrete-type (quasi-)norms.