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 \[ \partial_t^\alpha w(t,x) = \phi(\Delta) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, \] where $\partial_t^\alpha$ denotes the Caputo derivative of $\alpha \in (0,1)$ and $\phi(\Delta)$ is a nonlocal operator associated with a Bernstein function $\phi$. For all $p, q \in (1, \infty)$ and $\gamma \in \mathbb{R}$, we prove the estimate \begin{align*} &\left\| \partial_t^\alpha w \right\|_{L_q(0,T,\mu_2dt; H^{\phi,\gamma}_p(\mu_1))} + \left\| \phi(\Delta) w \right\|_{L_q(0,T,\mu_2dt; H^{\phi,\gamma}_p(\mu_1))} \\ &\qquad\leq C \left( \left\| h \right\|_{L_q(0,T,\mu_2dt; H^{\phi,\gamma}_p(\mu_1))} + \left\| w_0 \right\|_{N_{\alpha,p,\phi}} \right), \end{align*} where $\mu_1\in A_p(\mathbb{R}^d)$ and $\mu_2\in A_q(\mathbb{R})$ are Muckenhoupt weights, and $N_{\alpha,p,\phi}$ is a Banach space characterizing admissible initial data. In particular, when $\mu_2\equiv 1$ and $\alpha q>1$, $N_{\alpha,p,\phi}$ coincides with the weighted Besov space $B^{\phi,\gamma+2-\frac{2}{\alpha q}}_{p,q}(\mu_1)$. 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.