## Abstract

We present a weighted L_{q}(L_{p})-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation ∂_{t} ^{α}u(t,x)=Δu(t,x)+f(t,x),t>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,∞), w_{1}(x)∈A_{p} and w_{2}(t)∈A_{q}, ∫0∞(∫R^{d}|u_{xx}|^{p}w_{1}dx)^{q/p}w_{2}dt≤N∫0∞(∫R^{d}|f|^{p}w_{1}dx)^{q/p}w_{2}dt, 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.

Original language | English |
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Pages (from-to) | 3515-3550 |

Number of pages | 36 |

Journal | Journal of Differential Equations |

Volume | 269 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2020 Aug 5 |

### Bibliographical note

Funding Information:The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324).

Publisher Copyright:

© 2020 Elsevier Inc.

## Keywords

- Caputo fractional derivative
- Fractional diffusion-wave equation
- L(L)-theory
- Muckenhoupt A weights

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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