An optimal estimate for the norm of wavelet localization operators

TL;DR

This paper establishes an optimal norm estimate for wavelet localization operators with Cauchy wavelets, revealing different regimes based on Lebesgue norm ratios and characterizing optimal weights.

Contribution

It provides the first precise norm bounds for these operators under specific Lebesgue norm constraints and characterizes the optimal weight functions.

Findings

Multiple regimes depend on the ratio of Lebesgue norms.

Both constraints are active within a specific ratio interval.

Optimal weight functions are explicitly characterized.

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.