Weighted L_q(L_p)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

2019/11/18 by Beom-Seok Han, Han, Beom-Seok, Kyeong-Hun Kim +3 · 3 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1911.07437

openalex publication_date 2019/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a weighted L_q(L_p)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂_t^αu(t,x)=Δu(t,x) +f(t,x), tgt;0, x∈ ℝ^d. Here, α∈ (0,2) and ∂_t^α is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈ (1,∞), w₁(x)∈ A_p and w₂(t)∈ A_q, ∫^∞₀(∫_ℝ^d |u_xx|^p w₁ dx )^q/p w₂dt ≤ N ∫^∞₀(∫_ℝ^d |f|^p w₁ dx )^q/p w₂dt, 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.

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• Hölder Stable Recovery of the Source in Space-Time Fractional Wave Equations
• An approach for weighted mixed-norm estimates for parabolic equations with local and non-local time derivatives
• Muckenhoupt-weighted L_q(L_p) boundedness for time-space fractional nonlocal operators

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