Bounded linear Sobolev $L^{2,p}$ extension for arbitrary finite subsets of the plane when $1<p<2$

The open problem left by Drake, Fefferman, Ren and Skorobogatova in 'Sobolev extension in a simple case' (Advanced Nonlinear Studies 25 (2024), 260-278; arXiv:2402.11731) is the following. For $1<p<2$ and every finite set $E$ contained in a single line in $\mathbb{R}^2$, they constructed a bounded linear extension operator $T:L^{2,p}(E)\to L^{2,p}(\mathbb{R}^2)$ whose norm depends only on $p$, together with a sparse family of linear functionals, independent of the exponent $p$, characterizing the trace seminorm $\|f\|_{L^{2,p}(E)}$ up to constants depending only on $p$; each functional depends on the data at boundedly many points and the family has at most $C\#E$ elements, making the seminorm computable in linear time. Their Remark 1.1 states that these conclusions remain open for an arbitrary choice of finite $E\subset\mathbb{R}^2$. The problem is to prove or disprove, for every finite planar $E$ and every $1<p<2$: (a) existence of a bounded linear extension operator with constant depending only on $p$ (in the source's strength: $p$-independent and local), and (b) a sparse, $p$-independent functional characterization of the trace seminorm. Carruth and Israel (2025) proved an equivalence relating the planar problem on tree-associated sets to weighted Sobolev extension on finite trees with radially decaying weights (weights that decrease by a fixed factor along paths away from the root); it leaves both directions open and shows that a negative answer on trees would yield the first known negative answer to the general Sobolev extension problem.

Background

Let $L^{m,p}(\mathbb{R}^n)$ denote the homogeneous Sobolev space of real-valued functions on $\mathbb{R}^n$ whose distributional derivatives of order $m$ belong to $L^p$, with seminorm $\|F\|_{L^{m,p}(\mathbb{R}^n)}=(\sum_{|\alpha|=m}\int_{\mathbb{R}^n}|\partial^\alpha F|^p)^{1/p}$, that is, the $L^p$-norm of the $m$-th order derivatives. The Sobolev (Whitney-type) extension problem for finite data asks: given a finite set $E\subset\mathbb{R}^n$ and values $f:E\to\mathbb{R}$, understand the trace seminorm $\|f\|_{L^{m,p}(E)}=\inf\{\|F\|_{L^{m,p}(\mathbb{R}^n)}: F\in L^{m,p}(\mathbb{R}^n),\ F=f \text{ on } E\}$, and construct a bounded linear extension operator: a linear map $T$ from the space of functions on $E$ to $L^{m,p}(\mathbb{R}^n)$ with $Tf=f$ on $E$ and $\|Tf\|_{L^{m,p}(\mathbb{R}^n)}\le C\|f\|_{L^{m,p}(E)}$ for a constant $C$ depending only on $m,n,p$ and not on $E$. Linearity with such a uniform bound is what makes the theory operational: it underlies algorithms that compute the order of magnitude of the trace seminorm, and extensions of the data, in $O(N)$ operations for $N=\#E$.

This program is well understood in the supercritical regime $p>n$, where Sobolev embedding makes every $F\in L^{m,p}(\mathbb{R}^n)$ continuous so restrictions to arbitrary sets are meaningful. Israel (2013) settled the planar case $L^{2,p}(\mathbb{R}^2)$ for $p>2$, and Fefferman, Israel and Luli (2014) proved in general that for $p>n$ there is a bounded linear extension operator together with linear functionals $\ell_1,\dots,\ell_{\nu_{\max}}$ whose $p$-sum is equivalent to the trace seminorm, with $\nu_{\max}$ growing only linearly in $N$ and each $\ell_\nu$ depending on the data at boundedly many points; Drake (2023) developed the corresponding approximate extension theory for data measures. This is the 'fitting a smooth function to data' program.

Below the critical exponent these techniques break down, and the first nontrivial case is the planar one: $L^{2,p}(\mathbb{R}^2)$ with $1<p<2$. In this range point traces are still meaningful, because the embedding chain '$\nabla^2 F\in L^p$ implies $\nabla F\in L^{2p/(2-p)}$ implies $F$ is Hölder continuous of exponent $2-2/p>0$' makes every $F\in L^{2,p}(\mathbb{R}^2)$ continuous. At $p=1$ this fails and point values are not defined; at $p=2$ the Hilbert space structure of $L^{2,2}(\mathbb{R}^2)$ makes a norm-preserving linear extension immediate via orthogonal projection; so $1<p<2$ is precisely the open range. The analytic difficulty is that the trace seminorm is no longer local: when two points $z_1,z_2\in E$ near a square of the decomposition of side length $\delta$ are much closer together than $\delta$, their difference quotient $(f(z_2)-f(z_1))/(x_2-x_1)$ no longer approximates the gradient on that square, and the approximating difference quotient must instead involve a point $z^*$ of $E$ much farther away; relating every region of the plane to such far-away points, and summing the resulting estimates over all squares without unbounded overlaps, becomes the core obstacle. The standard proof machinery is a Calderón-Zygmund decomposition, a Whitney-type partition of a square containing $E$ into dyadic squares (squares whose side lengths are powers of two and which nest across scales) whose side length is comparable to their distance from $E$, with the interpolant assembled as a partition-of-unity blend of affine polynomials fitted locally to the data.

Drake, Fefferman, Ren and Skorobogatova (2024) settled the simplest nontrivial geometry, assuming $E$ lies on a single line. Their key new device is a grouping $G_P(z^*)$: for each point $z^*\in E$, the Calderón-Zygmund squares whose local difference quotient must involve the far-away point $z^*$ are collected into a single subcollection $G_P(z^*)$ whose rigid structure restores the overlap control lost when $1<p<2$. They obtain two linked conclusions, which together give a linear, norm-efficient and computable extension theory for collinear data: (Theorem 1) a linear $T:L^{2,p}(E)\to L^{2,p}(\mathbb{R}^2)$ with $Tf=f$ on $E$ and $\|Tf\|_{L^{2,p}(\mathbb{R}^2)}\le C_p\|f\|_{L^{2,p}(E)}$, where $T$ can be taken independent of $p$ and local, in that $Tf(z)$ is determined by the values of $f$ at at most $C$ points of $E$; and (Theorem 2) positive weights $\lambda_\nu$ and linear functionals $\ell_\nu$, independent of $p$, with $\nu_{\max}\le CN$, each $\ell_\nu(f)$ determined by the values of $f$ at at most $C$ points of $E$, such that $(\sum_\nu \lambda_\nu|\ell_\nu(f)|^p)^{1/p}$ is comparable to $\|f\|_{L^{2,p}(E)}$ up to constants depending only on $p$. The functional list is what makes the trace seminorm, the very quantity the operator is measured against, computable in $O(N)$ operations.

For non-collinear $E$ almost nothing is known in this range. Carruth and Israel (2024) constructed a bounded linear extension operator for one exceptional fractal set $E$ lying on the union of two closely spaced parallel lines, by reducing the problem to weighted Sobolev extension on a tree, where a theorem of Fefferman and Klartag (2023) on radially symmetric binary trees (trees in which all vertices at the same distance from the root carry the same weight) applies. Carruth and Israel (2025) then proved that for finite rooted trees with radially decaying weights (edge weights that decrease by at least a fixed factor along every path away from the root) the planar problem on an associated finite set $E$ is equivalent to weighted Sobolev extension on the tree, showing in particular that a negative answer on trees would give a negative answer in the plane, which would be the first known negative answer to the general Sobolev extension problem. The case of an arbitrary finite $E\subset\mathbb{R}^2$ is open in both directions.

Problem Statement

Fix $1<p<2$ and let $E\subset\mathbb{R}^2$ be an arbitrary finite set, not assumed to lie on a line. For $f:E\to\mathbb{R}$ define the trace seminorm $\|f\|_{L^{2,p}(E)}=\inf\{\|F\|_{L^{2,p}(\mathbb{R}^2)}: F\in L^{2,p}(\mathbb{R}^2),\ F=f \text{ on } E\}$. Drake, Fefferman, Ren and Skorobogatova proved, for $E$ contained in a single line, the existence of a bounded linear extension operator and of a sparse functional characterization of the trace seminorm, and stated (Remark 1.1) that these conclusions remain open for an arbitrary choice of finite $E\subset\mathbb{R}^2$. The problem is to settle, for every finite $E\subset\mathbb{R}^2$ and every $1<p<2$, the following two linked conclusions.

(a) Does there exist a bounded linear extension operator $T:L^{2,p}(E)\to L^{2,p}(\mathbb{R}^2)$, that is, a linear map with $Tf=f$ on $E$ and $\|Tf\|_{L^{2,p}(\mathbb{R}^2)}\le C_p\|f\|_{L^{2,p}(E)}$ for all $f$, where $C_p$ depends only on $p$ and not on $E$ or $\#E$; and, in the strength proved for collinear $E$, can $T$ in addition be taken independent of $p$, and local, in the sense that $Tf(z)$ is determined by the values of $f$ at at most $C$ points of $E$ for an absolute constant $C$?

(b) Does the trace seminorm admit a sparse, $p$-independent functional representation: do there exist positive weights $\lambda_1,\dots,\lambda_{\nu_{\max}}$ and linear functionals $\ell_1,\dots,\ell_{\nu_{\max}}$ on the space of functions on $E$, independent of $p$, with $\nu_{\max}\le C\#E$ and each $\ell_\nu(f)$ determined by the values of $f$ at at most $C$ points of $E$, such that $(\sum_{\nu=1}^{\nu_{\max}}\lambda_\nu|\ell_\nu(f)|^p)^{1/p}$ is comparable to $\|f\|_{L^{2,p}(E)}$ up to constants depending only on $p$?

Both conclusions are known when $E$ is contained in a line; the source paper highlights the case of $E$ contained in a union of two closely spaced parallel lines as the first geometry where substantial new issues arise. In the negative direction, the problem asks for a finite set $E$ for which no bounded linear extension operator with constant depending only on $p$ can be taken both independent of $p$ and local, or for which no $p$-independent sparse characterization of the trace seminorm with constants depending only on $p$ exists.

References

1. Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova, 'Sobolev extension in a simple case', Advanced Nonlinear Studies 25(2):260-278 (2024). DOI: 10.1515/ans-2023-0132, https://doi.org/10.1515/ans-2023-0132 (author preprint arXiv:2402.11731, https://arxiv.org/abs/2402.11731)
2. Jacob Carruth, Arie Israel, 'An example related to Whitney's extension problem for L^{2,p}(R^2) when 1<p<2', Advanced Nonlinear Studies 25(2):350-363 (2024). DOI: 10.1515/ans-2023-0126, https://doi.org/10.1515/ans-2023-0126 (author preprint arXiv:2312.07642, https://arxiv.org/abs/2312.07642)
3. Jacob Carruth, Arie Israel, 'The Sobolev extension problem on trees and in the plane', Advanced Nonlinear Studies 25(1):11-33 (2025). DOI: 10.1515/ans-2023-0158, https://doi.org/10.1515/ans-2023-0158 (author preprint arXiv:2406.12097, https://arxiv.org/abs/2406.12097)
4. Charles Fefferman, Bo'az Klartag, 'Linear extension operators for Sobolev spaces on radially symmetric binary trees', Advanced Nonlinear Studies 23(1), article 20220075 (2023). DOI: 10.1515/ans-2022-0075, https://doi.org/10.1515/ans-2022-0075
5. Arie Israel, 'A bounded linear extension operator for L^{2,p}(R^2)', Annals of Mathematics (2) 178(1):183-230 (2013). DOI: 10.4007/annals.2013.178.1.3, https://doi.org/10.4007/annals.2013.178.1.3
6. Charles L. Fefferman, Arie Israel, Garving K. Luli, 'Sobolev extension by linear operators', Journal of the American Mathematical Society 27(1):69-145 (2014). DOI: 10.1090/S0894-0347-2013-00763-8, https://doi.org/10.1090/S0894-0347-2013-00763-8
7. Marjorie K. Drake, 'Approximate extension in Sobolev space', Advances in Mathematics 420, article 108999 (2023). DOI: 10.1016/j.aim.2023.108999, https://doi.org/10.1016/j.aim.2023.108999
8. Pavel Shvartsman, Nahum Zobin, 'On planar Sobolev L_p^m-extension domains', Advances in Mathematics 287:237-346 (2016). DOI: 10.1016/j.aim.2015.08.031, https://doi.org/10.1016/j.aim.2015.08.031
9. Han Li, 'The existence of a bounded linear extension operator for L^{s,p}(R^n) when n/p < {s}', arXiv:2410.07367 (2024), https://arxiv.org/abs/2410.07367