7 research outputs found
Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators
This paper extends a class of degenerate elliptic operators for which
hypoellipticity requires more than a logarithmic gain of derivatives of a
solution in every direction. Work of Hoshiro and Morimoto in late 80s
characterized a necessity of a super-logarithmic gain of derivatives for
hypoellipticity of a sum of a degenerate operator and some non-degenerate
operators like Laplacian. The operators we consider are similar, but more
general. We examine operators of the form , where
is one-dimensional and may itself vanish. The argument of the paper is
based on spectral projections, analysis of a spectral differential equation and
interpolation between standard and operator-adapted derivatives. Unlike prior
results in the literature, our results do not require explicit analytic
construction in the non-degenerate direction. In fact, our result allows
non-analytic and even non-smooth coefficients for the non-degenerate part.Comment: Accepted for publication at the Journal of Mathematical Analysis and
Applications. 23 page