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

Abstract

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem {partial derivative(t)u(t, x) = psi(t, -i del)u(t, x) + f (t, x), (t, x) is an element of(0, T) x R-d, u(0, x) = 0, x is an element of R-d, (0.1) in (Muckenhoupt) weighted L-p-spaces with time-measurable pseudo-differential operators psi(t,-i del)u(t, x) := F-1 [psi(t, center dot)F[u](t, center dot)] (x). (0.2) More precisely, we find sufficient conditions of the symbol psi(t, xi) (especially depending on the smoothness of the symbol with respect to xi) to guarantee that equation (0.1) is well-posed in (Muckenhoupt) weighted L-p-spaces. Here the symbol psi(t, xi) is merely measurable with respect to t, and the sufficient smoothness of psi(t, xi) with respect to xi 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), integral(T)(0) integral(Rd) vertical bar(-Delta)(gamma/2)u(t, x)vertical bar(p)(t(2) + vertical bar x vertical bar(2))(alpha/2)dxdt (0.3) <= N integral(T)(0) integral(Rd) vertical bar f(t, x)vertical bar(p)(t(2) + vertical bar x vertical bar(2))(alpha/2)dxdt and integral(T)(0) (integral(Rd) vertical bar(-Delta)(gamma/2)u(t, x)vertical bar(p)vertical bar x vertical bar)(alpha/2)dx)(q/p) t(alpha 1)dt (0.4) <= N integral(T)(0) (integral(Rd) vertical bar f(t, x)vertical bar(p)vertical bar x vertical bar(alpha/2)dx)(q/p) t(alpha 1)dt, where p, q is an element of(1, infinity), -d - 1 < alpha < (d + 1)(p - 1), -1 < alpha(1) < q - 1, -d < alpha(2) < d(p - 1), and gamma is the order of the operator psi(t, -i del).