Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators
Yong Zhen Yang, Yong Zhou
TL;DR
The paper addresses weighted mixed-norm estimates for time-space fractional evolution equations driven by a nonlocal operator $φ(Δ)$ with Caputo time derivative $\partial_t^α$, under Muckenhoupt weights. It introduces a robust framework combining Marcinkiewicz multiplier theory for $φ(Δ)$, sharp-function techniques, and Fefferman–Stein inequalities to derive $L_q((0,T); L_p(μ_1))$ bounds in the presence of $μ_1∈A_p$ and $μ_2∈A_q$, and it provides a precise initial-data characterization via weighted Besov spaces $B^{φ,(W\circ k^{\star})^{1/q}}_{p,q}(μ_1)$. The main results establish existence, uniqueness, and a priori estimates for solutions in weighted spaces $\mathcal{H}^{α,φ,γ+2}_{p,q}(μ_1,μ_2,T)$, and they include a specialization to the unweighted case that recovers classical Besov-type initial-data spaces when $αq>1$. By unifying and generalizing previous weighted $L_q(L_p)$ theories, the work significantly broadens analytic tools for TSFEs with general Bernstein function generators and nonlocal dynamics. The results have potential applications in modeling anomalous diffusion and memory effects in heterogeneous media where weights reflect spatial-temporal inhomogeneities.
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.