Parabolic BMO estimates for pseudo-differential operators of arbitrary order

Abstract

In this article we prove the BMO-L-infinity estimate parallel to(-Delta)(gamma/2)u parallel to(BMO(Rd+1)<=) N parallel to partial derivative/partial derivative tu - A(t) u parallel to(L infinity (Rd+1)), for all u is an element of C-G(infinity) (Rd+1) for a wide class of pseudo-differential operators A(1) of order gamma is an element of (0, infinity). The coefficients of A(t) are assumed to be merely measurable in time variable. As an application to the equation partial derivative/partial derivative t u=A(t) u +f, t is an element of R we prove that for any u is an element of C-G(infinity) (Rd+1) parallel to u(t)parallel to(Lp)(Rd+1) + parallel to(-Delta)(gamma/2) u parallel to L-p(Rd+1)<= N parallel to u(t) - A(t)u parallel to L-p(Rd+1), where p is an element of (1, infinity) and the constant N is independent of u.