Degenerate elliptic operators as regularizers

Abstract:

The spaces ${\mathcal {K}_{mk}}$, introduced in the Nehari Volume of Journal d’Analyse Mathématique, for nonnegative integer values of $m$ and arbitrary real values of $k$ are extended to negative values of $m$. The extension is consistent with the equivalence $\parallel {\zeta ^j}u{\parallel _{m,k}}\sim \parallel u{\parallel _{m,k - j}}$, the inequality $\parallel {D^\alpha }u{\parallel _{m,k}} \leqslant {\text {const}}\parallel u{\parallel _{m + |\alpha |,k + |\alpha |}}$, and the generalized Cauchy-Schwarz inequality $|\langle {u,v} \rangle | \leqslant \parallel u {\parallel _{m,k}}\parallel v\parallel _{ - m, - k}$. (Here $\langle u, \upsilon \rangle$ is the ${L_2}$ scalar product.) There exists a second order degenerate elliptic operator which maps ${\mathcal {K}_{m,k}} 1 - 1$ onto ${\mathcal {K}_{m - 2,k}}$. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc. 28 (1981), 226.)