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

Schlag, W.

doi:10.1090/S0002-9939-06-08621-7

A remark on Littlewood-Paley theory for the distorted Fourier transform
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Proc. Amer. Math. Soc. 135 (2007), 437-451 Request permission

Abstract:

We consider the classical theorems of Mikhlin and Littlewood-Paley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator $-\Delta +V$. We show that for such operators which display a zero energy resonance the full range $1<p< \infty$ in the Mikhlin theorem cannot be obtained: in the radial, three-dimensional case it shrinks to $\frac 32<p<3$.

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