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 ${\partial }_t^{\alpha }$ denotes the Caputo derivative of $\alpha \in (0,1)$ and $\varphi (\Delta )$ is a nonlocal operator associated with a Bernstein function $\varphi$. For all $p,q\in (1,\infty )$ and $\gamma \in \mathbb{R}$, 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 ${\mu }_1\in A_p({\mathbb{R}}^d)$ and ${\mu }_2\in A_q(\mathbb{R})$ are Muckenhoupt weights, and $N_{\alpha ,p,\varphi }$ is a Banach space characterizing admissible initial data. In particular, when ${\mu }_2\equiv 1$ and $\alpha q>1$, $N_{\alpha ,p,\varphi }$ coincides with the weighted Besov space $B_{p,q}^{\varphi ,\gamma +2-\frac{2}{\alpha q}}({\mu }_1)$. 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.

37 pages