Improving Estimates for Differential Operators in Two Independent Variables
Humio Suzuki · Publications of the Research Institute for Mathematical Sciences · 1969
IntroductionLet P(x 9 D) be a differential operator of order m with C°°c oefficients defined in an open subset O, of R n .We shall say that an improving estimate for P holds in H, if the following condition is fulfilled :For every compact set K in fl, there exists a positive number P>0 such that where \\u\\ s is the norm of the space H s [_!, §2.4].In this paper we shall study conditions for the validity of improving estimates, when the number of independent variables is two.We denote the principal symbol of P(x, D) by p(x, £).define the kth commutator Cp(x, £) of p(x, £) by induction :where [^, ^] is the Poisson bracket of p and q, f -9^,-9^z-9 1/ Let k p (x, ^) denote the first value of k for which Cp(x, ?)=j=0.If c^(jr, f) vanishes for all values of k, we define ^(tf, f) to be oo.Our main result is the following