Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators

Abstract

We develop a weighted mixed-norm $L_q(L_p)$-estimates for solutions to fractional evolution equations of the form \[ \partial_t^αw(t,x) = ϕ(Δ) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, \] where $\partial_t^α$ denotes the Caputo derivative of $α\in (0,1)$ and $ϕ(Δ)$ is a nonlocal operator associated with a Bernstein function $ϕ$. For all $p, q \in (1, \infty)$ and $γ\in \mathbb{R}$, we prove the estimate \begin{align*} &\left\| \partial_t^αw \right\|_{L_q(0,T,μ_2dt; H^{ϕ,γ}_p(μ_1))} + \left\| ϕ(Δ) w \right\|_{L_q(0,T,μ_2dt; H^{ϕ,γ}_p(μ_1))} \\ &\qquad\leq C \left( \left\| h \right\|_{L_q(0,T,μ_2dt; H^{ϕ,γ}_p(μ_1))} + \left\| w_0 \right\|_{N_{α,p,ϕ}} \right), \end{align*} where $μ_1\in A_p(\mathbb{R}^d)$ and $μ_2\in A_q(\mathbb{R})$ are Muckenhoupt weights, and $N_{α,p,ϕ}$ is a Banach space characterizing admissible initial data. In particular, when $μ_2\equiv 1$ and $αq>1$, $N_{α,p,ϕ}$ coincides with the weighted Besov space $B^{ϕ,γ+2-\frac{2}{αq}}_{p,q}(μ_1)$. The analysis employs tools from harmonic analysis, including the Fefferman--Stein inequality, Hardy-Littlewood maximal estimates in weighted mixed-norm spaces, and sharp function methods for bounding solution operators. These results extend and unify previous work by K.~H.~Kim et al, providing a general analytic framework for weighted $L_q(L_p)$-theory of time-space nonlocal evolution equations.