Weighted L_q(L_p)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

We present a weighted L_q(L_p)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂_t ^αu(t,x)=Δu(t,x)+f(t,x),t>0,x∈R^d. Here, α∈(0,2) and ∂_t ^α is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈(1,∞), w₁(x)∈A_p and w₂(t)∈A_q, ∫0∞(∫R^d|u_xx|^pw₁dx)^q/pw₂dt≤N∫0∞(∫R^d|f|^pw₁dx)^q/pw₂dt, where A_p is the class of Muckenhoupt A_p weights. Our approach is based on the sharp function estimates of the derivatives of solutions.