Characterization of minimax-optimal regularized differentiation operators

Characterize all minimax optimal operators for regularized spectral differentiation on Sobolev spaces H^{s,p}(R^n), rather than providing only the sufficient admissibility conditions established for Fourier multiplier operators.

Background

The paper introduces directly verifiable Mikhlin and low-frequency approximation conditions for families of Fourier multipliers and proves that the associated regularized differentiation operators attain the minimax rate E^{|b1|/s}b4^{1-|b1|/s} for 1<p<a5 and 0c|b1|<s. These conditions are sufficient for optimality but are not asserted to be necessary.

The unresolved problem is to identify a complete class of minimax-optimal operators. The difficulty is especially pronounced for general L^p spaces, where bounded operators need not admit a simple Fourier-multiplier characterization and practical symbol criteria such as the Mikhlin conditions provide only convenient sufficient tests.