## Abstract

When studying non-symmetric nonlocal operators on R^{d}: Lf(x)=∫R^{d}(f(x+z)−f(x)−∇f(x)⋅z1_{{|z|⩽1}})[Formula presented]dz, where 0<α<2, d⩾1, and κ(x,z) is a function on R^{d}×R^{d} that is bounded between two positive constants, it is customary to assume that κ(x,z) is symmetric in z. In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption κ(x,z)=κ(x,−z). In fact, we allow the kernel κ to be time-dependent and x→κ(t,x,z) to be only locally β-Hölder continuous with Hölder constant possibly growing at a polynomial rate in |z|. We also derive gradient estimate when β∈(0∨(1−α),1) as well as fractional derivative estimate of order θ∈(0,(α+β)∧2) for the heat kernel. Moreover, when α∈(1,2), drift perturbation of the time-dependent non-local operator L_{t} with drift in Kato's class is also studied in this paper. As an application, when κ(x,z)=κ(z) does not depend on x, we show the boundedness of nonlocal Riesz's transformation: for any p>2d/(d+α), ‖L^{1/2}f‖_{p}≍‖Γ(f)^{1/2}‖_{p}, where Γ(f):=[Formula presented]L(f^{2})−fLf is the carré du champ operator associated with L, and L^{1/2} is the square root operator of L defined by using Bochner's subordination. Here ≍ means that both sides are comparable up to a constant multiple.

Original language | English |
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Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 465 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Sept 2018 |

Externally published | Yes |

## Keywords

- Heat kernel estimates
- Levi's method
- Non-symmetric nonlocal operator
- Riesz's transform