Traces of Sobolev functions and higher integrability

TL;DR

The paper addresses how interior higher integrability in ${\rm L}^{q}$ improves boundary traces of ${\rm W}^{1,p}$-Sobolev functions, proving a sharp trace characterization: the trace of ${\rm W}^{1,p}\cap{\rm L}^{q}$ on a Lipschitz boundary is exactly ${\rm W}^{1-\frac{1}{p},p}\cap{\rm L}^{r}$, with $r=1+q\left(1-\frac{1}{p}\right)$. A key feature is the introduction of a nonlinear extension or lifting operator, built from a Poisson extension and a magnitude-dependent cutoff, which realizes the trace and yields the optimal $r$-exponent; the proof covers $1<p<n$ and $p^{*}<q\le\infty$, with a separate, elementary treatment of the $p=1$ case following classical approaches. The paper also discusses variations for operator-adapted spaces $W^{\mathbb{A},p}$ under $\mathbb{C}$-ellipticity, establishing analogous trace results and extending the framework to broader differential-operator settings. Overall, the work sharpens the link between interior integrability and boundary regularity, yielding precise trace spaces and constructive lifting mechanisms applicable to PDE boundary-value problems and related function-space theory.

Abstract

We give a sharp characterization of how additional integrability in the interior improves the integrability of boundary traces of $\mathrm{W}^{1,p}$-Sobolev functions. The optimality of our results relies on a novel nonlinear extension or lifting operator.

Figures (2)

• Figure 1: (Non-)collapse of the support of $u$. If $|v(x',x_{n})|\to\infty$ sufficiently fast as $x_{n}\searrow 0$, then it might happen that $u(x',0)=0$, see (i). If this happens at the points of too large a set, \ref{['eq:itrainsdowninafrica']} might not necessarily be fulfilled. If $v$ is bounded, as is the case in the setting of Step 1 and 2, this scenario cannot occur; see (ii). The conclusion of the continuity argument from Step 3 thus can be interpreted as (i) not happening too often for general $({\dot{{\rm W}}}^{1-1/p,p}\cap\mathrm{L}^{r})$-functions $f\colon\mathbb{R}^{n-1}\to\mathbb{R}$; see also Remark \ref{['rem:choice']}.
• Figure 2: The geometric setting in Section \ref{['sec:variations']}. Far away from $\mathbb{R}^{n-1}$, $u$ is left unchanged (see term $\mathcal{T}_{j}^{(1)}u$ in \ref{['eq:replacementsequence']}) while, close to $\mathbb{R}^{n-1}$, it is locally replaced by a weighted sum of projections onto the finite dimensional null spaces (see term $\mathcal{T}_{j}^{(2)}u$ in \ref{['eq:replacementsequence']}).

Theorems & Definitions (19)

• Theorem 1.1: Trace space of ${\rm W}^{1,p}\cap\mathrm{L}^{q}$
• Theorem 1.2: $p=1$, Müller Mueller
• Lemma 2.1
• Lemma 2.2: Properties of the Poisson extension
• Lemma 2.3
• proof
• Remark 2.4
• proof : Proof of Theorem \ref{['thm:main']}, Case $1<p<p^{*}<q<\infty$
• proof : Proof of Theorem \ref{['thm:main']}, case $q=\infty$
• Remark 2.5