The omitted details in Fefferman's linear interpolation and extrapolation operators: an explicit bounded-depth construction for finite sets and the $m=1$ case of the Banach-limit passage

Fefferman's 2005 paper 'Interpolation and extrapolation of smooth functions by linear operators' proved that for every finite set $E$ in $\mathbb{R}^n$ with a tolerance function $\sigma$ there is a linear operator $T$ from functions on $E$ to $C^m(\mathbb{R}^n)$ that reproduces any function admitting a near-optimal $C^m$ interpolant, and that for arbitrary $E$ there is a bounded linear operator on the trace space $C^{m-1,1}(E,\sigma)$. Two parts of the argument were asserted but not written out: (A) the operator of the finite-set theorem 'may be taken to have bounded depth' — each point of $\mathbb{R}^n$ has a neighborhood on which $Tf$ depends only on the restriction of $f$ to an a-priori bounded number of points of $E$ — with the details omitted; (B) the passage from finite sets to the trace space via Banach limits was carried out only under the assumption $m \ge 2$, the case $m = 1$ being left to the reader. The adjacent research question raised in the same introduction — the Brudnyi–Shvartsman conjecture on existence of bounded linear extension operators — was resolved affirmatively by Fefferman in 2007, and the successor literature never publishes the omitted construction: the 2007 structure paper quotes the bounded-depth result back to the 2005 paper without proof, and the 2009 paper 'Extension of $C^{m,\omega}$-Smooth Functions by Linear Operators' states a generalized bounded-depth refinement for finite sets (its Theorem 8) but only in sketch form, in a jet-based depth formulation that controls only the jet at the points of $E$ rather than the function on a neighborhood, and without an explicit depth constant. The surviving open problem is to produce the complete written account: a written-out construction of a bounded-depth operator for the finite-set interpolation theorem, with an a-priori depth bound $N(m,n)$ depending only on $m$ and $n$, and a complete proof of the Lipschitz case ($m=1$) of the trace-space theorem.

Background

For integers $m, n \ge 1$, let $C^{m,1}(\mathbb{R}^n)$ denote the space of functions on $\mathbb{R}^n$ whose derivatives of order $m$ are Lipschitz, and for a subset $E \subset \mathbb{R}^n$ let $C^{m,1}(E)$ be the space of restrictions to $E$ of such functions — a trace space. A classical problem going back to H. Whitney (1934) asks how to decide whether a given function $f: E \to \mathbb{R}$ is the restriction of a function of prescribed smoothness on $\mathbb{R}^n$, and whether the extension can be produced by a bounded linear operator. Brudnyi and Shvartsman conjectured in the 1990s that the trace space of $C^m$ on any subset of $\mathbb{R}^n$ admits a bounded linear extension operator; their 1997 paper settled special cases, and the conjecture was later proven in full by Fefferman's Annals paper 'C^m extension by linear operators' (2007). The 2005 Revista Matemática Iberoamericana paper (Charles Fefferman, 'Interpolation and extrapolation of smooth functions by linear operators') proves two refinements in the same circle of ideas. Throughout that paper, $\|F\|_{C^m(\mathbb{R}^n)}$ denotes $\sup_{x \in \mathbb{R}^n} \max_{|\beta| \le m} |\partial^\beta F(x)|$, the maximum running over multi-indices $\beta$, and $\|F\|_{C^{m-1,1}(\mathbb{R}^n)}$ denotes $\max_{|\beta| \le m-1}\big(\sup_{x} |\partial^\beta F(x)| + \sup_{x \neq y} \frac{|\partial^\beta F(x) - \partial^\beta F(y)|}{|x - y|}\big)$. Theorem 1 (finite-set form): given a finite set $E \subset \mathbb{R}^n$ and a tolerance $\sigma: E \to [0,\infty)$, there is a linear map $T$ from the functions on $E$ into $C^m(\mathbb{R}^n)$ such that whenever $f$ admits an interpolant $F \in C^m(\mathbb{R}^n)$ with $\|F\|_{C^m} \le 1$ and $|F(x)-f(x)| \le \sigma(x)$ on $E$, the constructed $Tf$ satisfies $\|Tf\|_{C^m} \le A$ and $|Tf(x)-f(x)| \le A\,\sigma(x)$ on $E$, with $A$ depending only on $m$ and $n$. Theorem 2 (trace-space form): for arbitrary $E$ and $\sigma$, there is a linear map $T: C^{m-1,1}(E,\sigma) \to C^{m-1,1}(\mathbb{R}^n)$ — where $C^{m-1,1}(E,\sigma)$ is the Banach space of functions $f: E \to \mathbb{R}$ for which there exist $M > 0$ and $F \in C^{m-1,1}(\mathbb{R}^n)$ with $\|F\|_{C^{m-1,1}(\mathbb{R}^n)} \le M$ and $|F(x)-f(x)| \le M\sigma(x)$ for all $x \in E$, normed by the infimum of all such $M$ — such that every $f$ with $\|f\|_{C^{m-1,1}(E,\sigma)} \le 1$ satisfies $\|Tf\|_{C^{m-1,1}(\mathbb{R}^n)} \le A$ and $|Tf(x)-f(x)| \le A\sigma(x)$ on $E$; by linearity this is equivalent to $\|Tf\| \le A\|f\|$ and $|Tf(x)-f(x)| \le A\sigma(x)\|f\|$ for all $f$, again with $A = A(m,n)$. Here a jet $J_y(F)$ is the Taylor polynomial of $F$ of degree $m-1$ at $y$, and the proof of Theorem 1 (which builds on Fefferman's 'A sharp form of Whitney's extension theorem', Ann. of Math. 2005) hinges on producing, for each point $y$, a suitable jet that depends linearly on $f$; Theorem 2 is then deduced by applying Theorem 1 to larger and larger finite subsets of $E$ and passing to a Banach limit — a linear functional on bounded functions of a discrete parameter, obtained from the Hahn–Banach theorem, whose value always lies between the liminf and the limsup. Two structural points were left unwritten. First, the operator of Theorem 1 'may be taken to have bounded depth': every point of $\mathbb{R}^n$ has a neighborhood $U$ such that $Tf|_U$ depends only on $f|_S$ for some $S \subset E$ whose cardinality is bounded a priori in terms of $m$ and $n$; the author states that this 'follows without difficulty from our proof of Theorem 1, but we omit the details'. A later paper by the same author ('The Structure of Linear Extension Operators for $C^m$', Rev. Mat. Iberoam. 2007) defines an analogous pointwise notion of depth ($Tf(x_0)$ a fixed linear combination of at most $s$ point evaluations at points of $E$, with the points and coefficients independent of $f$ — a slightly weakened form of the 2005 neighborhood notion), restates the finite-set bounded-depth result while citing it back to the 2005 paper without proof, proves that finite depth is in general impossible for infinite sets (a countable compact $E \subset \mathbb{R}^2$ on which $C^1(E)$ admits no finite-depth extension operator), and introduces 'breadth' — dependence on boundedly many one-point differential operators, i.e. linear functionals on the trace space given by evaluating a derivative of order at most $m$ at a single point, well defined because such an evaluation vanishes on any smooth function that is zero on $E$ — as the structural property that does survive for arbitrary $E$, but again only sketches the modified proof. Second, the Banach-limit argument establishing Theorem 2 is written under the assumption $m \ge 2$, 'leaving to the reader the task of modifying our arguments for the case $m = 1$' — the Lipschitz case, in which the target space is $C^{0,1}(\mathbb{R}^n)$ and the smoothness of the limit function must be obtained without the Taylor-expansion step used for $m \ge 2$. A successor paper by the same author ('Extension of $C^{m,\omega}$-Smooth Functions by Linear Operators', Rev. Mat. Iberoam. 2009) proves the analogous results with a general regular modulus of continuity in place of Lipschitz — a modulus of continuity is an increasing function $\omega$ with $\omega(0) = 0$ that governs how fast the $m$-th derivatives may vary, and 'regular' is a mild technical hypothesis on it. That paper proves both the pure-extension case ($\sigma \equiv 0$, its Theorem 2) and the $\sigma$-tolerance interpolation case (its Theorem 4, the 2005 paper having established its case $\omega(t) = t$), and in its final section it comes closest of all successors to the omitted details: its Theorem 8 states that for finite $E$ the operators of its Theorems 2–5 may be taken to have bounded depth in its own sense — for each $x \in E$ the Taylor jet $J_x(Tf)$ of order $m$ at $x$ is determined, as $f$ varies, by the values of $f$ on some set of at most $k$ points of $E$, and on every ball whose distance to $E$ exceeds its own radius the whole restriction $Tf|_B$ is determined by the values of $f$ on some set of at most $k$ points, with $k$ depending only on $m$ and $n$. The proof of Theorem 8, however, is a sketch: its key ingredient (its Theorem 9, a refinement of the paper's main structure theorem to so-called $k$-admissible maps — maps whose jet at every point factors through boundedly many linear functionals of the data) is described as a 'straightforward adaptation' and is not written out; the step establishing determination on balls that avoid $E$ is asserted from the classical Whitney extension theorem without proof; no explicit value or bound for $k$ is given; and the depth notion is not the 2005 neighborhood notion — it controls only the jet at the points of $E$, not the function on a neighborhood of those points, so it does not by itself establish the 2005 assertion. That paper also fixes $m \ge 1$ in its own numbering, which corresponds to $m \ge 2$ in the numbering of the 2005 trace-space theorem, so it does not supply the $m = 1$ account either; indeed it remarks that introducing the Banach limit into the argument would destroy the admissibility (locality) property, and it poses the infinite-set analogue of Theorem 8 as an open question.

Problem Statement

Produce the complete, fully written-out account of the two constructions that Fefferman's paper 'Interpolation and extrapolation of smooth functions by linear operators' (Rev. Mat. Iberoam. 21 (2005), 313–348) asserts but omits, as stated below. Throughout, $m, n \ge 1$ are integers, all norms are those of Section 1 of that paper, and every constant named $A$ or $N$ below must depend only on $m$ and $n$ — never on the set $E$, the tolerance $\sigma$, or the data $f$. (A) Bounded depth for the finite-set operator. For every finite set $E \subset \mathbb{R}^n$ and every tolerance $\sigma: E \to [0,\infty)$, construct explicitly a linear operator $T = T(E,\sigma)$ from $C(E)$, the vector space of real-valued functions on $E$, into $C^m(\mathbb{R}^n)$, satisfying the conclusion of Theorem 1 of that paper — whenever $f \in C(E)$ admits an interpolant $F \in C^m(\mathbb{R}^n)$ with $\|F\|_{C^m(\mathbb{R}^n)} \le 1$ and $|F(x) - f(x)| \le \sigma(x)$ for all $x \in E$, the constructed $Tf$ satisfies $\|Tf\|_{C^m(\mathbb{R}^n)} \le A$ and $|Tf(x) - f(x)| \le A\,\sigma(x)$ for all $x \in E$ — together with the additional property of bounded depth: every point of $\mathbb{R}^n$ has a neighborhood $U$ such that $Tf|_U$ depends only on $f|_S$ for a subset $S \subset E$ with $\#S \le N(m,n)$, where $N(m,n)$ is a finite a-priori bound; the neighborhood $U$ and the set $S$ may depend on $E$ and $\sigma$ but must not depend on $f$. The construction must be proved to have bounded depth and to satisfy the norm and interpolation bounds of $Tf$ (the 2005 paper asserts that this 'follows without difficulty from our proof of Theorem 1, but we omit the details'; the related 2007 paper of the same author quotes the result without proving it; and the same author's 2009 paper 'Extension of $C^{m,\omega}$-Smooth Functions by Linear Operators' states a closely related finite-set bounded-depth refinement in its Theorem 8, but only sketches the proof — its key ingredient is asserted to be a 'straightforward adaptation' and is not written out — in a jet-based depth formulation that controls only the jet at the points of $E$ rather than the function on a neighborhood, and without an explicit depth constant). The target is any finite a-priori bound $N(m,n)$ depending only on $m$ and $n$; determining how small the depth can be — the optimal $N(m,n)$ — is a natural quantitative refinement on which the literature appears to be silent (the source asserts only that $\#S$ is 'bounded a-priori in terms of m and n'). (B) The case $m = 1$ of Theorem 2. Establish Theorem 2 for $m = 1$ in complete detail. Here $C^{0,1}(E,\sigma)$ is the Banach space of functions $f: E \to \mathbb{R}$ for which there exist $M > 0$ and $F \in C^{0,1}(\mathbb{R}^n)$ with $\|F\|_{C^{0,1}(\mathbb{R}^n)} \le M$ and $|F(x) - f(x)| \le M\sigma(x)$ for all $x \in E$, normed by the infimum of all such $M$, where $\|F\|_{C^{0,1}(\mathbb{R}^n)} = \sup_{x \in \mathbb{R}^n}|F(x)| + \sup_{x \neq y,\, x,y \in \mathbb{R}^n} |F(x) - F(y)|/|x - y|$; the target is a linear map $T: C^{0,1}(E,\sigma) \to C^{0,1}(\mathbb{R}^n)$ such that every $f$ with $\|f\|_{C^{0,1}(E,\sigma)} \le 1$ satisfies $\|Tf\|_{C^{0,1}(\mathbb{R}^n)} \le A$ and $|Tf(x) - f(x)| \le A\,\sigma(x)$ for all $x \in E$ — equivalently, by linearity, $\|Tf\|_{C^{0,1}(\mathbb{R}^n)} \le A\|f\|_{C^{0,1}(E,\sigma)}$ and $|Tf(x) - f(x)| \le A\,\sigma(x)\|f\|_{C^{0,1}(E,\sigma)}$ for all $f$ — with $A$ depending only on $n$. The source carries out Section 17 only under the assumption $m \ge 2$, 'leaving to the reader the task of modifying our arguments for the case $m = 1$'. Both components together constitute one objective: a publication-quality completion of the two omitted details of the 2005 paper, faithful to its definitions, norm conventions, and dimensional-constant requirements. A valid answer must not weaken the hypotheses or conclusions: in particular, it may not drop the tolerance function $\sigma$ (an account of pure extension only, $\sigma = 0$, does not suffice), restrict the class of sets $E$, let the constants depend on $E$ or $\sigma$, replace the 2005 neighborhood-based depth property by a different locality notion — the pointwise depth of the 2007 structure paper, or the jet-based depth of the 2009 paper, which controls only the jet at the points of $E$ together with the behavior on balls avoiding $E$ — or treat only some pairs $(m,n)$.

References

1. Charles Fefferman, "Interpolation and extrapolation of smooth functions by linear operators", Revista Matemática Iberoamericana 21 (2005), no. 1, 313–348. DOI: 10.4171/rmi/424, https://doi.org/10.4171/rmi/424 (source paper; open-access full text at https://ems.press/content/serial-article-files/38123)
2. Charles Fefferman, "The Structure of Linear Extension Operators for $C^m$", Revista Matemática Iberoamericana 23 (2007), no. 1, 269–280. DOI: 10.4171/rmi/495, https://doi.org/10.4171/rmi/495 (open-access full text at https://ems.press/content/serial-article-files/38193)
3. Charles Fefferman, "$C^m$ extension by linear operators", Annals of Mathematics (2) 166 (2007), no. 3, 779–835. DOI: 10.4007/annals.2007.166.779, https://doi.org/10.4007/annals.2007.166.779 (publisher page: https://annals.math.princeton.edu/2007/166-3/p04)
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