Weighted estimates for fractional powers of partial differential operators

Abstract:

It is shown that fractional powers defined by the wave polynomial $P(\xi ) = \xi _{^1}^2 + \cdots + \xi _p^2 - \xi _{p + 1}^2 - \cdots - \xi _n^2$, defined in terms of Fourier transforms by $\widehat {{T^\lambda }f} = {\left | {P(\xi )} \right |^\lambda }\hat f$, are in the Bernstein subalgebra of functions with integrable Fourier transforms for $\lambda > (n - 1)/2$, provided $f \in C_c^m$ with $m$ large enough. The proof uses embedding theorems for Besov spaces and Stein’s theorem on interpolation of analytic families of operators.