An optimal estimate for the norm of wavelet localization operators
Federico Riccardi
TL;DR
The paper derives sharp, regime-dependent bounds for the norm of the wavelet localization operator $L_{F,\beta}$ under dual Lebesgue constraints on the weight $F$, i.e., $F\in L^p(\mathbb{C}_+,d\nu)\cap L^q(\mathbb{C}_+,d\nu)$ with $\|F\|_{L^p}\le A$ and $\|F\|_{L^q}\le B$. The authors identify two inactive-constraint regimes with explicit extremizers and then address the remaining regime via a variational reduction to a maximization over distribution functions, yielding a unique maximizer expressed through $u(t)=4\pi\max\{(\lambda_1 t^{p-1}+\lambda_2 t^{q-1})^{-{1/(2\beta+1)}}-1,0\}$ and corresponding $\lambda_1,\lambda_2>0$ determined by the constraints. They prove that the optimal bounds take the form $\|L_{F,\beta}\| \le \dfrac{2\beta}{(4\pi)^{1/p}} \sigma_p^{\kappa_p} A$ (or the $L^q$- analogue) in the inactive regimes, with equality for weights supported on hyperbolic discs; otherwise the bound is obtained by the integral $\int_0^{\infty} G(v(t))\,dt$ with $v$ the distribution function of $|F|$, maximized by $u$. The results extend sharp norm estimates from time-frequency localization and connect to optimal weight shapes via a variational framework, with potential transfers to Bergman spaces and Toeplitz operators.
Abstract
In this paper we prove an optimal estimate for the norm of wavelet localization operators with Cauchy wavelet and weight functions that satisfy two constraints on different Lebesgue norms. We prove that multiple regimes arise according to the ratio of these norms: if this ratio belongs to a fixed interval (which depends on the Lebesgue exponents) then both constraints are active, while outside this interval one of the constraint is inactive. Furthermore, we characterize optimal weight functions.