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

Abstract

We present a weighted Lq(Lp)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation∂tαu(t,x)=Δu(t,x)+f(t,x),t>0,x∈Rd. Here, α∈(0,2) and ∂tα is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈(1,∞), w1(x)∈Ap and w2(t)∈Aq,∫0∞(∫Rd|uxx|pw1dx)q/pw2dt≤N∫0∞(∫Rd|f|pw1dx)q/pw2dt, where Ap is the class of Muckenhoupt Ap weights. Our approach is based on the sharp function estimates of the derivatives of solutions.