In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$-\partial _tu + \textrm{div}(A\nabla u)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>+</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on a Lipschitz cylinder <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {O}\times \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is solvable for some <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p\in (1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> under the assumption that the matrix A is elliptic, has bounded and measurable coefficients, and that its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition).We prove that for some <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p_0>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> the Regularity problem is solvable in the range <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(1,p_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small.In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.

The $L^p$ Regularity Problem for Parabolic Operators

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发表日期：

2026年7月25日

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Martin Dindoš,
Linhan Li,
Jill Pipher

摘要

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial _tu + \textrm{div}(A\nabla u)=0$ $- \partial_{t} u + div (A \nabla u) = 0$ on a Lipschitz cylinder $\mathcal {O}\times \mathbb {R}$ $O \times R$ is solvable for some $p\in (1,\infty )$ $p \in (1, \infty)$ under the assumption that the matrix A is elliptic, has bounded and measurable coefficients, and that its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition).

We prove that for some $p_0>1$ $p_{0} > 1$ the Regularity problem is solvable in the range $(1,p_0)$ $(1, p_{0})$ . We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small.

In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.

The \(L^p\) Regularity Problem for Parabolic Operators

摘要