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A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space

Jae-Hwan Choi, Ildoo Kim, Jin Bong Lee

TL;DR

This work develops a regularity theory for homogeneous initial-value problems governed by time-measurable pseudo-differential operators in weighted mixed-norm spaces. By combining Littlewood–Paley analysis with Laplace-transform techniques, the authors establish existence, uniqueness, and maximal regularity under general temporal weights beyond the Muckenhoupt class, with initial data in variable-smoothness Besov spaces. The core contributions include sharp, frequency-local kernel bounds for the solution operator, a robust mean-oscillation framework, and a precise equivalence between the time-dependent operator and the fractional Laplacian under regularity and ellipticity assumptions. The results extend trace and inhomogeneous solvability to broad weighted settings, enabling applications to nonzero initial data and forcing terms, and providing a flexible toolkit for parabolic-type problems with nonstandard temporal weights.

Abstract

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order.

A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space

TL;DR

This work develops a regularity theory for homogeneous initial-value problems governed by time-measurable pseudo-differential operators in weighted mixed-norm spaces. By combining Littlewood–Paley analysis with Laplace-transform techniques, the authors establish existence, uniqueness, and maximal regularity under general temporal weights beyond the Muckenhoupt class, with initial data in variable-smoothness Besov spaces. The core contributions include sharp, frequency-local kernel bounds for the solution operator, a robust mean-oscillation framework, and a precise equivalence between the time-dependent operator and the fractional Laplacian under regularity and ellipticity assumptions. The results extend trace and inhomogeneous solvability to broad weighted settings, enabling applications to nonzero initial data and forcing terms, and providing a flexible toolkit for parabolic-type problems with nonstandard temporal weights.

Abstract

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order.
Paper Structure (9 sections, 25 theorems, 308 equations)

This paper contains 9 sections, 25 theorems, 308 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Examples and Applications
  4. Proof of Theorem \ref{['22.12.27.16.53']}
  5. Proof of Proposition \ref{['prop:maximal esti']}
  6. Estimates on fundamental solutions
  7. Proof of Proposition \ref{['prop:maximal esti']}
  8. Weighted multiplier and Littlewood--Paley theorem
  9. Properties of function spaces

Key Result

Theorem 1.2

Let $T \in (0,\infty)$, $p,\nu\in(1,\infty)$, $q\in(0,\infty)$, $w\in A_{\nu}(\mathbb{R})$ and $w'\in A_p(\mathbb{R}^d)$. Assume that $a^{ij}(t)$ is a measurable function on $(0,T)$ for all $i,j \in \{1,\ldots,d\}$ and there exist positive constants $\kappa$ and $M$ such that Suppose that $u_0\in\mathcal{S}'(\mathbb{R}^d)$ satisfies where $W(\lambda):=\int_0^{\lambda}w(t)\mathrm{d}t$. Then the i

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7: Solution
  • Remark 2.8
  • ...and 54 more