Parabolic BMO estimates for pseudo-differential operators of arbitrary order

In this article we prove the BMO-L∞L∞ estimate‖(−Δ)γ/2u‖BMO(Rd+1)≤N‖∂∂tu−A(t)u‖L∞(Rd+1),∀u∈Cc∞(Rd+1) for a wide class of pseudo-differential operators A(t)A(t) of order γ∈(0,∞)γ∈(0,∞). The coefficients of A(t)A(t) are assumed to be merely measurable in time variable. As an application to the equation∂∂tu=A(t)u+f,t∈R we prove that for any u∈Cc∞(Rd+1)‖ut‖Lp(Rd+1)+‖(−Δ)γ/2u‖Lp(Rd+1)≤N‖ut−A(t)u‖Lp(Rd+1),‖ut‖Lp(Rd+1)+‖(−Δ)γ/2u‖Lp(Rd+1)≤N‖ut−A(t)u‖Lp(Rd+1), where p∈(1,∞)p∈(1,∞) and the constant N is independent of u.