A remark on Littlewood-Paley theory for the distorted Fourier transform

TL;DR

This paper extends the Littlewood-Paley theory to the distorted Fourier transform associated with Schrödinger operators, demonstrating the Mikhlin multiplier theorem's validity in a specific range for radial functions with potential resonance.

Contribution

It establishes the Mikhlin multiplier theorem for the distorted Fourier transform in the range (3/2,3) for radial functions, considering the impact of zero energy resonance.

Findings

Mikhlin multiplier theorem holds for radial functions in the specified range

The Littlewood-Paley theorem is valid within the same range

Resonance at zero energy restricts the theorem's applicability

Abstract

We show that the usual Mikhlin multiplier theorem relative to the distorted Fourier transform holds in the range (3/2,3) at least for radial functions and potentials. The restricted range is due to the fact that the Schroedinger operator which gives rise to the distorted Fourier transform may have a zero energy resonance. Consequently, the Littlewood-Paley theorem is shown also only for the range (3/2,3).