Linear extension operators for Sobolev spaces on weighted trees: the endpoint cases $p=1$ and $p=\infty$, the inhomogeneous space $W^{1,p}(T)$, non-radially-symmetric weights, and the planar $\dot W^{2,p}(\mathbb{R}^2)$ extension problem for $1<p<2$

Fefferman and Klartag (Advanced Nonlinear Studies, 2023) proved that on the full binary tree of height $N$ with positive edge weights depending only on the depth, there exists, for every $1<p<\infty$, a linear extension operator from the leaves to the whole tree whose weighted homogeneous Sobolev $\dot W^{1,p}$ norm is bounded by a constant depending only on $p$; the operator is a harmonic extension with respect to a Markov kernel tuned to $p$ and the weights. They explicitly left open the endpoint cases $p=1$ and $p=\infty$ and the analogous statement for the inhomogeneous space $W^{1,p}(T)$, and they conjectured that for edge weights not depending only on depth, in the range $p\neq2$ (at $p=2$ the standard harmonic extension has norm one even for arbitrary weights), no linear extension operator with a $p$-only norm bound exists. They also posed the driving planar question: for $1<p<2$, does every set $E\subset\mathbb{R}^2$ admit a bounded linear extension operator $\dot W^{2,p}(E)\to\dot W^{2,p}(\mathbb{R}^2)$ with norm depending only on $p$? These four questions share one objective, completing the existence theory of bounded linear Sobolev extension operators on weighted trees and the linked planar extension problem, and all four remain open in their stated generality: work in 2023-2025 constructed planar extension operators only for special classes of $E$ (finite subsets of a line; one explicit fractal-type set) and proved that for finite trees with radially decaying weights the tree and planar extension problems are equivalent, while explicitly recording that nothing is known for trees that are not perfect or whose weights are not radially symmetric.

Background

Homogeneous Sobolev spaces. For $1\le p\le\infty$ and an integer $m\ge1$, the homogeneous Sobolev space $\dot W^{m,p}(\mathbb{R}^n)$ (also written $L^{m,p}(\mathbb{R}^n)$) consists of the functions whose distributional derivatives of order $m$ belong to $L^p$, equipped with the seminorm $\|F\|_{\dot W^{m,p}}=\|\nabla^m F\|_{L^p}$; the inhomogeneous space $W^{m,p}$ augments this seminorm by the $L^p$-norm of the function itself. When $p>n/m$, functions in these spaces are continuous, so their restrictions to arbitrary subsets are well defined.

The Whitney extension problem for Sobolev spaces. For a set $E\subset\mathbb{R}^n$ (classically a finite set) define the trace seminorm $\|f\|_{\dot W^{m,p}(E)}=\inf\{\|F\|_{\dot W^{m,p}(\mathbb{R}^n)}: F|_E=f\}$. The problem, a Sobolev analogue of Whitney's classical extension theorem for jets of smooth functions, asks whether there exists a bounded linear extension operator $T$ with $(Tf)|_E=f$ and $\|Tf\|_{\dot W^{m,p}(\mathbb{R}^n)}\le C\|f\|_{\dot W^{m,p}(E)}$ for a constant $C=C(m,n,p)$ independent of $E$. Israel proved this for $L^{2,p}(\mathbb{R}^2)$ when $p>2$, and Fefferman, Israel and Luli proved it in full generality for $p>n$. Shvartsman had earlier obtained the trace characterization for the inhomogeneous $W^{1,p}$ spaces on closed subsets of $\mathbb{R}^n$. In the complementary (subcritical) range $n/m<p\le n$, whose first nontrivial instance is $L^{2,p}(\mathbb{R}^2)$ with $1<p<2$, the general question is open. In that range $L^{2,p}(\mathbb{R}^2)$ embeds into the continuous functions but not into $C^1$, so derivative (jet) data on $E$ are not available and the Whitney-style machinery of the $p>n$ theory does not extend.

The tree model. Fefferman and Klartag introduced a discrete model for the subcritical planar problem. Let $V=\bigcup_{k=0}^{N}\{0,1\}^k$ be the vertex set of the full binary tree of height $N$: a vertex is a binary string of length at most $N$, the root is the empty string, the leaves are $\partial V=\{0,1\}^N$, and $\pi x$ denotes the parent of $x$ (the string $x$ with its last bit deleted). Given positive radial edge weights $W_1,\dots,W_N$, where $W_k$ is the weight of every edge joining a vertex of depth $k$ to its parent, define for $F:V\to\mathbb{R}$

$\|F\|_{\dot W^{1,p}(V)}=\Big(\sum_{k=1}^{N}W_k\sum_{x\in\{0,1\}^k}|F(x)-F(\pi x)|^p\Big)^{1/p},$

and $\|f\|_{\dot W^{1,p}(\partial V)}=\inf\{\|F\|_{\dot W^{1,p}(V)}: F|_{\partial V}=f\}$ for $f:\partial V\to\mathbb{R}$. A linear extension operator is a linear map $H:\mathbb{R}^{\partial V}\to\mathbb{R}^V$ with $(Hf)|_{\partial V}=f$ for every $f$; the goal is a bound $\|Hf\|_{\dot W^{1,p}(V)}\le C_p\|f\|_{\dot W^{1,p}(\partial V)}$ with $C_p$ independent of $N$ and of the weights. For $p=2$ the harmonic extension operator, which fills in each vertex with the expected leaf value under the standard symmetric random walk started at that vertex and stopped at the leaves, has norm one. The even simpler averaging operator, which assigns to each vertex the average of the leaf values beneath it, is bounded only under additional hypotheses on the weights (Björn, Björn, Gill and Shanmugalingam, in their geometric analysis on Cantor sets and trees) and fails, for instance, when $W_k=2^{-k}$.

The Fefferman-Klartag theorem. For every $1<p<\infty$ and every radial weight sequence there is a linear extension operator with $\bar C_p\le 4p^{1/p}q^{1/q}\big(1+\max\{(p-1)^{-1/p},(q-1)^{-1/q}\}\big)$, where $q=p/(p-1)$, so the bound depends only on $p$. The operator is again a harmonic extension, but for a Markov kernel (the transition law of a random walk) tuned to $p$ and the weights: a walk started at a vertex of depth $s$ reaches a leaf before visiting depth $s-1$ with probability $q_s=(2^sW_s)^{-1/(p-1)}/\sum_{k=s}^{N}(2^kW_k)^{-1/(p-1)}$. The norm estimate is reduced to a one-dimensional weighted $\ell^p$ operator on the depth levels, bounded via the classical discrete Muckenhoupt two-weight condition for Hardy's inequality (Muckenhoupt's standard characterization of the pairs of weights for which the one-dimensional weighted averaging operator is bounded on $\ell^p$). The authors stress that the kernel genuinely depends on $p$: for some weight sequences, the operator built for one value $p_0$ is not uniformly bounded for any other $p\in(1,\infty)$.

Why trees matter for the plane. The tree problem arose directly from the planar $\dot W^{2,p}$ extension problem: Fefferman and Klartag note that for a particular class of sets $E\subset\mathbb{R}^2$ the planar problem reduces to their theorem. Carruth and Israel then used this reduction to construct a bounded linear extension operator for $L^{2,p}$ on an explicit fractal-type planar set, and later proved a converse: for any finite tree with radially decaying weights (weights satisfying $W_v\le\varepsilon W_{\pi v}$ for small enough $\varepsilon>0$, not assumed radially symmetric, on a tree not assumed perfect) there is a set $E\subset\mathbb{R}^2$ for which the tree extension problem and the planar $L^{2,p}$ extension problem are equivalent, in that a bounded linear extension operator exists for one if and only if for the other, with comparable operator norms. The tree questions are therefore not isolated curiosities: a negative answer for radially decaying weights would give the first known negative answer to the general Sobolev extension problem, while positive answers feed the planar program.

Problem Statement

The following four questions were left open by Fefferman and Klartag in their study of linear extension operators for Sobolev spaces on radially symmetric binary trees.

(1) Endpoint exponents. For $p=1$ and for $p=\infty$, does there exist a finite constant $C_p$ such that for every height $N$ and every choice of positive radial weights $W_1,\dots,W_N$ on the full binary tree there is a linear extension operator $H:\mathbb{R}^{\partial V}\to\mathbb{R}^V$ with $(Hf)|_{\partial V}=f$ and $\|Hf\|_{\dot W^{1,p}(V)}\le C_p\|f\|_{\dot W^{1,p}(\partial V)}$ for all $f:\partial V\to\mathbb{R}$? The known construction and its Muckenhoupt-based proof are tied to $1<p<\infty$ and degenerate at both endpoints, so the existence question is genuinely open there. The source's closing remark phrases this question for general weighted trees $T$ while the paper's theorem concerns radially symmetric binary trees; this target is pinned to the radially weighted full binary tree formulated above, and settling that pinned baseline settles the target, the broader weighted-tree reading remaining a natural extension beyond the pinned scope. The two endpoint exponents are logically independent questions.

(2) Inhomogeneous norm. Does the analogous statement hold when the homogeneous seminorm is replaced by an inhomogeneous norm on the tree, i.e. the seminorm above augmented by an $L^p$-term $\big(\sum_{x\in V}\mu(x)\,|F(x)|^p\big)^{1/p}$ over the vertices for the plain counting measure $\mu\equiv1$ on $V$ (the source asks only for the analog of its homogeneous result and does not pin the vertex measure, so this target fixes that convention), together with the corresponding inhomogeneous trace norm on the leaves, again with a constant depending only on $p$? This target is pinned to the counting measure on $V$, and settling that pinned convention settles the target; the wider reading in which $\mu$ is a weighted counting measure associated with the edge weights remains a natural extension beyond the pinned scope.

(3) Non-radially-symmetric weights. Resolve the Fefferman-Klartag conjecture for $p\neq2$: prove that for general positive edge weights, not required to depend only on the depth, there is no linear extension operator whose norm is bounded by a function of $p$ alone; or refute the conjecture by constructing, for arbitrary edge weights, a linear extension operator with such a bound. The restriction $p\neq2$ is part of the conjecture as posed in the source: for arbitrary (not necessarily radial) weights at $p=2$, the standard harmonic extension operator already has norm one, whereas the source records that "for $p\neq2$ the situation seems subtle". (The conjecture is closely related to the Banach-space question of which subspaces of $\ell^p_n$, i.e. $\mathbb{R}^n$ with the $\ell^p$-norm, admit linear projections of norm bounded independently of the dimension.)

(4) Planar problem. For $1<p<2$ and an arbitrary set $E\subset\mathbb{R}^2$ (restrictions are meaningful for any $E$ here, since $L^{2,p}(\mathbb{R}^2)$ functions are continuous in this range), does there exist a bounded linear extension operator $T:\dot W^{2,p}(E)\to\dot W^{2,p}(\mathbb{R}^2)$ with $(Tf)|_E=f$ and $\|Tf\|_{\dot W^{2,p}(\mathbb{R}^2)}\le C_p\|f\|_{\dot W^{2,p}(E)}$ for a constant $C_p$ independent of $E$? The answer is known to be affirmative for $p>2$, and for $1<p<2$ only for special classes of sets $E$.

These four targets share one scientific objective: to complete the existence theory of bounded linear Sobolev extension operators on weighted trees and, through the established tree-to-plane reductions, the associated planar extension problem in the first open subcritical case.

References

1. Charles Fefferman and Bo'az Klartag, "Linear extension operators for Sobolev spaces on radially symmetric binary trees", Advanced Nonlinear Studies 23 (2023), no. 1, article 20220075. DOI: 10.1515/ans-2022-0075, https://doi.org/10.1515/ans-2022-0075 (author manuscript: arXiv:2301.13792, https://arxiv.org/abs/2301.13792)
2. Arie Israel, "A bounded linear extension operator for $L^{2,p}(\mathbb{R}^2)$", Annals of Mathematics (2) 178 (2013), no. 1, 183-230. DOI: 10.4007/annals.2013.178.1.3, https://doi.org/10.4007/annals.2013.178.1.3
3. Charles L. Fefferman, Arie Israel, and Garving K. Luli, "Sobolev extension by linear operators", Journal of the American Mathematical Society 27 (2014), no. 1, 69-145. DOI: 10.1090/s0894-0347-2013-00763-8, https://doi.org/10.1090/s0894-0347-2013-00763-8
4. Jacob Carruth and Arie Israel, "An example related to Whitney's extension problem for $L^{2,p}(\mathbb{R}^2)$ when $1<p<2$", Advanced Nonlinear Studies 25 (2025), no. 2, 350-363. DOI: 10.1515/ans-2023-0126, https://doi.org/10.1515/ans-2023-0126 (preprint: arXiv:2312.07642, https://arxiv.org/abs/2312.07642)
5. Marjorie Drake, Charles Fefferman, Kevin Ren, and Anna Skorobogatova, "Sobolev extension in a simple case", Advanced Nonlinear Studies 25 (2025), no. 2, 260-278. DOI: 10.1515/ans-2023-0132, https://doi.org/10.1515/ans-2023-0132 (preprint: arXiv:2402.11731, https://arxiv.org/abs/2402.11731)
6. Jacob Carruth and Arie Israel, "The Sobolev extension problem on trees and in the plane", Advanced Nonlinear Studies 25 (2025), no. 1, 11-33. DOI: 10.1515/ans-2023-0158, https://doi.org/10.1515/ans-2023-0158 (preprint: arXiv:2406.12097, https://arxiv.org/abs/2406.12097)
7. Anders Björn, Jana Björn, James T. Gill, and Nageswari Shanmugalingam, "Geometric analysis on Cantor sets and trees", Journal für die reine und angewandte Mathematik 725 (2017), 63-114. DOI: 10.1515/crelle-2014-0099, https://doi.org/10.1515/crelle-2014-0099
8. Benjamin Muckenhoupt, "Hardy's inequality with weights", Studia Mathematica 44 (1972), 31-38. DOI: 10.4064/sm-44-1-31-38, https://doi.org/10.4064/sm-44-1-31-38
9. Pavel Shvartsman, "Sobolev $W^{1}_p$-spaces on closed subsets of $\mathbb{R}^n$", Advances in Mathematics 220 (2009), no. 6, 1842-1922. DOI: 10.1016/j.aim.2008.09.020, https://doi.org/10.1016/j.aim.2008.09.020