Weighted L-q(L-p)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

Abstract

We present a weighted L-q(L-p)-theory (p, q is an element of (1, infinity)) with Muckenhoupt weights for the equation partial derivative(alpha)(t)u(t, x) = Delta u(t, x) + f(t, x), t > 0, x is an element of R-d. Here, alpha is an element of (0, 2) and partial derivative(alpha)(t) is the Caputo fractional derivative of order alpha. In particular we prove that for any p, q is an element of (1, infinity), w(1) (X) is an element of A(p) and w(2) (t) is an element of A(q), integral(infinity)(0)(integral(Rd) vertical bar u(xx)vertical bar(p) w(1)dx)(q/p) w(2)dt <= N integral(infinity)(0)(integral(Rd) vertical bar f vertical bar(p) w(1)dx)(q/p) w(2)dt, 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. (C) 2020 Elsevier Inc. All rights reserved.