An L_q(L_p)-theory for diffusion equations with space-time nonlocal operators

We present an L_q(L_p)-theory for the equation ∂_t^αu=ϕ(Δ)u+f,t>0,x∈R^d;u(0,⋅)=u₀. Here p,q>1, α∈(0,1), ∂_t^α is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ₀∈(0,1] and c>0 such that [Formula presented] We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove ‖|∂_t^αu|+|u|+|ϕ(Δ)u|‖_Lq([0,T];Lp)≤N(‖f‖_Lq([0,T];Lp)+‖u₀‖_{Bp,q^ϕ,2−2/αq}), where B_p,q^ϕ,2−2/αq is a modified Besov space on R^d related to ϕ. Our approach is based on BMO estimate for p=q and vector-valued Calderón-Zygmund theorem for p≠q. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.