Circumventing the Gaussian multiplier regularity restriction

Determine whether the admissibility restriction s-|b1|c=2 for the Gaussian Fourier multiplier m_{R,b1}(be)=exp(-(|be|/R)^2) in H^{s,2}(R^n) can be circumvented.

Background

The paper verifies that the Gaussian multiplier satisfies the admissibility conditions in the Hilbert-space case p=2 when the Sobolev regularity gap satisfies 0c=s-|b1|c=2, using the estimate 1-e^{-x^2}c=x^2. This restriction arises because the Gaussian approximation error is of quadratic order at low frequencies, whereas admissibility requires approximation of order s-|b1|.

The authors do not establish whether another argument, a modification of the admissibility framework, or a sharper property of the Gaussian multiplier would allow admissibility beyond this regularity range. Thus, the unresolved issue concerns extending the Gaussian example to cases with s-|b1|>2.