A weighted L_p -regularity theory for parabolic partial differential equations with time-measurable pseudo-differential operators

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem {∂tu(t,x)=ψ(t,-i∇)u(t,x)+f(t,x),(t,x)∈(0,T)×Rd,u(0,x)=0,x∈Rd, in (Muckenhoupt) weighted L_p -spaces with time-measurable pseudo-differential operators ψ(t,-i∇)u(t,x):=F-1[ψ(t,·)F[u](t,·)](x). More precisely, we find sufficient conditions of the symbol ψ(t, ξ) (especially depending on the smoothness of the symbol with respect to ξ) to guarantee that equation (0.1) is well-posed in (Muckenhoupt) weighted L_p -spaces. Here the symbol ψ(t, ξ) is merely measurable with respect to t, and the sufficient smoothness of ψ(t, ξ) with respect to ξ is characterized by a property of each weight. In particular, we prove the existence of a positive constant N such that for any solution u to equation (0.1), ∫0T∫Rd|(-Δ)γ/2u(t,x)|p(t2+|x|2)α/2dxdt≤N∫0T∫Rd|f(t,x)|p(t2+|x|2)α/2dxdt and ∫0T(∫Rd|(-Δ)γ/2u(t,x)|p|x|α2dx)q/ptα1dt≤N∫0T(∫Rd|f(t,x)|p|x|α2dx)q/ptα1dt, where p, q∈ (1 , ∞) , - d- 1 < α< (d+ 1) (p- 1) , - 1 < α₁< q- 1 , - d< α₂< d(p- 1) , and γ is the order of the operator ψ(t, - i∇) .