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$L_p$-estimates for nonlocal equations with general Lévy measures

Hongjie Dong, Junhee Ryu

TL;DR

This work addresses $L_p$-type regularity for time-inhomogeneous nonlocal parabolic equations driven by general Lévy measures $\nu_t$ of order $\sigma\in(0,2)$, allowing highly singular kernels and no time regularity assumptions. It develops an analytic framework based on level-set arguments and a DR86-type maximal-function theory for general measures to prove continuity of the operators, a priori $L_p$ estimates, and unique solvability in unweighted spaces; in the 1D case with $\sigma\in(1,2)$, it further yields weighted mixed-norm solvability, while showing the weighted theory generally fails for higher dimensions or for $d=1$, $\sigma\in(0,1)$. The methodology avoids probabilistic representations and instead relies on tail/mean-oscillation controls and carefully constructed maximal functions tailored to general Lévy measures. The results extend prior nonlocal regularity theory to highly singular operators and remove temporal regularity requirements, broadening applicability to nonlocal PDEs with complex jump structures.

Abstract

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(σ)} \right) ν_t(dy), \end{equation*} where $ν_t$ is a general Lévy measure of order $σ\in(0,2)$. We allow this class of Lévy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $σ$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.

$L_p$-estimates for nonlocal equations with general Lévy measures

TL;DR

This work addresses -type regularity for time-inhomogeneous nonlocal parabolic equations driven by general Lévy measures of order , allowing highly singular kernels and no time regularity assumptions. It develops an analytic framework based on level-set arguments and a DR86-type maximal-function theory for general measures to prove continuity of the operators, a priori estimates, and unique solvability in unweighted spaces; in the 1D case with , it further yields weighted mixed-norm solvability, while showing the weighted theory generally fails for higher dimensions or for , . The methodology avoids probabilistic representations and instead relies on tail/mean-oscillation controls and carefully constructed maximal functions tailored to general Lévy measures. The results extend prior nonlocal regularity theory to highly singular operators and remove temporal regularity requirements, broadening applicability to nonlocal PDEs with complex jump structures.

Abstract

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(σ)} \right) ν_t(dy), \end{equation*} where is a general Lévy measure of order . We allow this class of Lévy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in spaces are established. We also demonstrate that, depending on the ranges of and , the operator can or cannot be treated in weighted mixed-norm spaces.
Paper Structure (8 sections, 20 theorems, 275 equations)

This paper contains 8 sections, 20 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Solvability of equations with general operators
  4. The case of $p=2$
  5. Auxiliary results
  6. Proof of Theorem \ref{['mainthm']}
  7. A special case in weighted spaces
  8. Miscellaneous lemmas

Key Result

Theorem 2.3

Let $d\geq1$, $\sigma\in(0,2)$, $T\in(0,\infty)$, $\lambda\geq0$, and $p\in(1,\infty)$. Suppose that $L_t$ and $\mathcal{L}_t$ satisfy Assumptions levy and upper, respectively. $(i)$ For each $t>0$, the operators $L_t$ and $\mathcal{L}_t$ are continuous from $H_{p}^{\sigma}(\mathbb{R}^d)$ to $L_p(\m where $v\in H_{p}^{\sigma}(\mathbb{R}^d)$. In particular, when $\sigma=1$, one can also consider $\

Theorems & Definitions (44)

  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 34 more