An example related to Whitney’s extension problem for L ^2,p (R²) when 1 < p < 2

1 Introduction

Let X ( R n ) be a space of continuous, real-valued functions on R n equipped with a norm (or seminorm) ‖ ⋅ ‖. For any subset Ω ⊂ R n , define the trace norm (or seminorm) of a continuous function f : Ω → R by

The trace space X ( Ω ) is then defined to be the space of continuous functions f : Ω → R with finite trace norm.

We say that a linear map T : X ( Ω ) → X ( R n ) is a linear extension operator for X ( Ω ) provided that (Tf)|_Ω = f for all f ∈ X ( Ω ) . We say that X ( Ω ) admits a bounded linear extension operator if there exists a constant C (determined only by the function space X ( R n ) ) such that there exists a bounded linear extension operator for X ( Ω ) with operator norm at most C. One formulation of the Whitney extension problem for X ( R n ) asks whether X ( Ω ) admits a bounded linear extension operator for every subset Ω ⊂ R n .

Let X ( R n ) = L m , p ( R n ) denote the (homogeneous) Sobolev space of all functions F : R n → R whose (distributional) partial derivatives of order m belong to L p ( R n ) , equipped with the seminorm ‖ F ‖ L m , p ( R n ) = ‖ ∇ m F ‖ L p ( R n ) . We assume that n m < p < ∞ , so that functions F ∈ L m , p ( R n ) are continuous – this ensures that the trace space L ^m,p (Ω) is well-defined for an arbitrary subset Ω ⊂ R n .

In Ref. [1], the second-named author, C. Fefferman, and G. K. Luli proved that L ^m,p (Ω) admits a bounded linear extension operator when p > n, with operator norm at most a constant C = C(m, n, p), for an arbitrary subset Ω ⊂ R n . When p is in the range m n < p ≤ n , however, it is unknown whether such an operator exists in general.

The first nontrivial case in this range is the space X = L 2 , p ( R 2 ) for 1 < p < 2 (when p = 2, it is easy to construct a bounded linear extension operator thanks to the Hilbert space structure). In this paper, we construct a bounded linear extension operator for L ^2,p (E) for a particular set E ⊂ R 2 to be defined below. This set E appears to be exceptional. For a variety of sets Ω ⊂ R 2 it is possible to construct a bounded linear extension operator for L ^2,p (Ω) by making certain analogies with the case p > 2. Our construction for L ^2,p (E), however, requires completely new ideas.

We now define the set E. We introduce a real number ɛ ∈ (0, 1/2). We assume that ɛ is smaller than some absolute constant. For an integer L ≥ 1, we define

The set E is of the form

where

Accordingly, E ₁ is a set of equispaced Δ-separated points on the line x ₂ = 0, and E ₂ is a set of points of separation ≥ Δ on the line x ₂ = Δ. For convenience, we assume that for every (x, Δ) ∈ E ₂ we have (x, 0) ∈ E ₁. We can ensure this by, e.g., taking ɛ = 1/N for a large, positive integer N.

In this paper, we prove the following theorem.

We now say a bit about the construction of the operator T. We fix a square Q 0 ⊂ R 2 containing E with diameter ∼ 1 . Given f : E → R , we will describe how to produce a function F ∈ L ^2,p (Q ⁰) satisfying F| _E = f and having the property that

for any G ∈ L 2 , p ( R 2 ) for which G| _E = f. Once we accomplish this, it is not difficult to deduce Theorem 1.

We first partition the square Q ⁰ into a Calderón–Zygmund decomposition CZ, which is a finite family of dyadic squares Q ⊂ Q ⁰ with pairwise disjoint interiors. Briefly, every Q ∈ CZ has sidelength δ _Q ≈ Δ + dist(Q, E) and satisfies #(1.1Q ∩ E) ≤ 1. In particular, the smallest squares in CZ have sidelength proportional to Δ and distance to E bounded by CΔ – see Section 3 for further details. Our function F ∈ L ^2,p (Q ⁰) takes the form

where { θ Q } Q ∈ CZ is a Whitney partition of unity subordinate to CZ – more precisely, (1) ∑ _Q θ _Q = 1 on Q ⁰, (2) supp(θ _Q ) ⊂ 1.1Q, and (3) | ∇ k θ Q | ≲ δ Q − k for each Q ∈ CZ – and each P _Q is an affine polynomial satisfying

Using (3) and the support condition on θ _Q , we see that the function F will satisfy F| _E = f. For each Q ∈ CZ, P _Q is of the form

where η _Q is a real number and L _Q is an affine polynomial satisfying ∂₂ L _Q = 0. Due to the structure of the set E ₁, our hand is essentially forced when it comes to choosing the L _Q ’s – specifically, we demand that each L _Q agrees with the function f at two suitably chosen points of the set E ₁. The bulk of the work, then, is in choosing the numbers { η Q } Q ∈ CZ to ensure a certain natural compatibility condition on the family { P Q } Q ∈ CZ , namely, the condition:

where Q ↔ Q′ denotes the property that 1.1Q ∩ 1.1Q′ ≠ ∅ for two squares Q, Q′ ∈ CZ (we say that Q “touches” Q′), and G ∈ L 2 , p ( R 2 ) is any function satisfying G| _E = f. Estimate (4) is the essential ingredient in the proof of the norm bound (1) for the function F defined in (2).

Roughly speaking, each η _Q corresponds to the average x ⁽²⁾-partial derivative of F on the square Q. Since the value of f near points of the set E ₂ is our only source of information about the x ⁽²⁾-partial derivative of F, we make use of a natural hierarchical clustering C of the set E ₂ to choose the η _Q ’s.

Elements C ∈ C correspond to the subsets of E ₂ of the form C ( s 1 , … , s k ) ≔ ( z , Δ ) : z = ∑ ℓ = 1 L s ℓ ε ℓ , s k + 1 , … , s L ∈ { − 1 , + 1 } for fixed (s ₁, …, s _k ) ∈ {−1,+1} ^k . The smallest clusters correspond to the 2 ^L different singleton sets in E ₂ (when k = L), and the largest cluster corresponds to C = E ₂ (when k = 0). Distinct clusters are “well-separated” in the sense that dist(C ₁, C ₂) ≫ max{diam(C ₁), diam(C ₂)} for C 1 , C 2 ∈ C , C ₁ ≠ C ₂.

The set of clusters C has the structure of a rooted binary tree of depth L. The clusters corresponding to the leaves of the tree are the singletons {z} (z ∈ E ₂), and E ₂ is the root of C .

To every square Q ∈ CZ, we associate a cluster C Q ∈ C . If 1.1Q ∩ E ₂ ≠ ∅ then Q is associated to the singleton cluster C _Q = {x _Q } where x _Q is a point of 1.1Q ∩ E ₂. To every other square Q we associate a cluster C Q ∈ C that is nearby to Q.

We then introduce real numbers { η C } C ∈ C and postulate that

For each z = (z ⁽¹⁾, Δ) ∈ E ₂, we define

This is natural, as for squares Q ∈ CZ satisfying 1.1Q ∩ E ₂ ≠ ∅ we will need to choose η Q = η { x Q } where x _Q ∈ 1.1Q ∩ E ₂, to ensure that P _Q satisfies (3).

Now that we have chosen ( η { z } ) z ∈ E 2 , we need to choose the remaining η _C so as to establish the compatibility condition (4) for the P _Q . We use the following theorem of C. Fefferman and B. Klartag, proved in [2].

Write ∂V ⊂ V to denote the set of leaves of V (i.e., ∂V = V _N ). Let R V and R ∂ V denote the sets of real-valued functions on V and ∂V, respectively.

Given weights w ₁, …, w _N > 0, define the L ^1,p (V)-seminorm by

Define the L ^1,p trace seminorm by

There exists a linear operator H : R ∂ V → R V with the following properties:

1. H is an extension operator, i.e.,

   H ϕ | ∂ V = ϕ for any ϕ ∈ R ∂ V .

2. For any ϕ ∈ R ∂ V , we have

   ‖ H ϕ ‖ L 1 , p ( V ) ≲ ‖ ϕ ‖ L 1 , p ( ∂ V ) .

We apply Theorem 2 to the binary tree C equipped with a natural set of edge weights, and choose the η _C by applying the extension operator H to the data { η { z } } z ∈ E 2 ; because the resulting choice of {η _C } minimizes the seminorm (6), one can show the resulting polynomials {P _Q } defined in terms of the {η _Q } (which are related to the {η _C } in (5)) will satisfy (4). As mentioned before, the norm bound (1) follows as a consequence of this condition.

Because the η _{z} depend linearly on f, and the η _C depend linearly on { η { z } } z ∈ E 2 , it is obvious that the P _Q depend linearly on f, and so, F depends linearly on f.

It is finally trivial to extend F : Q 0 → R to a function F ̃ : R 2 → R satisfying ‖ F ̃ ‖ L 2 , p ( R 2 ) ≤ C ‖ F ‖ L 2 , p ( Q 0 ) and F ̃ | Q 0 = F (in particular, F ̃ | E = f ). Since F ∈ L ^2,p (Q ⁰) is an extension of f satisfying the norm bound (1), we deduce that F ̃ ∈ L 2 , p ( R 2 ) is an extension of f satisfying ‖ F ̃ ‖ L 2 , p ( R 2 ) ≤ C ‖ G ‖ L 2 , p ( R 2 ) for any G ∈ L 2 , p ( R 2 ) with G| _E = f. Thus, we can set T f = F ̃ , and T is a bounded linear extension operator for L ^2,p (E).

This concludes our overview of the proof of Theorem 1.

We remark that when ɛ is bounded away from a certain “critical value” ɛ ₀ = (1/2)^1/(2−p), it is possible to prove Theorem 1 without invoking Theorem 2. In particular, one can construct an extension operator for L ^2,p (E) by making certain analogies with the case p > 2. When ɛ = ɛ ₀, however, we know of no such construction.

Throughout this paper we write C, C′, C″, … to denote positive constants. Such constants are allowed to depend only on p (but not on ɛ, L). We write C X , C X ′ , … for positive constants depending on some quantity X. For positive real numbers A, B we write A ≲ B if there exists a constant C such that A ≤ CB and A ≲ _X B if A ≤ C _X B. We write A ≈ B if A ≲ B and A ≲ B.

We thank Charles Fefferman, Anna Skorobogatova, and Ignacio Uriarte-Tuero, for helpful conversations. We also thank Pavel Shvartsman for suggesting that we use the Hardy–Littlewood maximal function; this greatly simplified some of the arguments. Finally, we thank the anonymous referee for helpful comments that led to the improvement of this article.

2 Preliminaries

For a (Lebesgue) measurable function F defined on a measurable set S ⊂ R 2 with |S| > 0, we write (F) _S ≔ |S|⁻¹∫ _S F dx. We begin by stating a couple of standard inequalities for Sobolev spaces. For details see, e.g., [3].

Given an annulus A = x ∈ R 2 : r ≤ | x − x 0 | ≤ R with inner radius r and outer radius R, the thickness ratio of A is defined to be the quantity R/r.

We use these inequalities to prove the following basic lemma.

Let B(z, r) denote the ball of radius r > 0 centered at z ∈ R 2 , and let M denote the uncentered Hardy–Littlewood maximal operator, i.e.,

Recall that M is a bounded operator from L q ( R 2 ) to L q ( R 2 ) for any 1 < q ≤ ∞ (see, e.g., [4]).

3 The CZ decomposition

We will work with squares in R 2 ; by this we mean an axis parallel square of the form Q = [a ₁, b ₁) × [a ₂, b ₂). We let δ _Q denote the sidelength of such a square Q. To bisect a square Q is to partition Q into squares Q ₁, Q ₂, Q ₃, Q ₄, where δ Q i = δ Q / 2 for each i = 1, 2, 3, 4. We refer to the Q _i as the children of Q.

We define a square Q ⁰ = [−4, 4) × [−4, 4); note that E ⊂ Q ⁰. A dyadic square Q is one that arises from repeated bisection of Q ⁰. Every dyadic square Q ≠ Q ⁰ is the child of some square Q′; we call Q′ the parent of Q and denote this by (Q)⁺ = Q′.

We say that two dyadic square Q, Q′ touch if (1.1Q ∩ 1.1Q′) ≠ ∅. We write Q ↔ Q′ to denote that Q touches Q′.

For any dyadic square Q, we define a collection CZ(Q), called the Calderón–Zygmund decomposition of Q, by setting

and

We write CZ = CZ(Q ⁰). Note that CZ ≠ {Q ⁰} because #(3Q ⁰ ∩ E) = #E ≥ 2. Then CZ is a partition of Q ⁰ into dyadic squares Q satisfying

To see this, observe that for any Q ∈ CZ we have #(3Q ⁺ ∩ E) ≥ 2 and 3Q ⁺ ⊂ 9Q. Since points of E are Δ-separated, we have δ _Q ≥Δ/9, as claimed.

We now summarize some basic properties of the collection CZ.

We omit the proof of Lemma 3, as this type of decomposition is standard in the literature; see, e.g., [5].

We say that Q ∈ CZ is a boundary square if 1.1Q ∩ ∂Q ⁰ ≠ ∅. Denote the set of boundary squares by ∂CZ.

Observe that every Q ∈ ∂CZ satisfies δ _Q ≥ 1. Indeed, this follows because E ⊂ [−1, 1] × [−1, 1], and if Q is a dyadic square intersecting the boundary of Q ⁰ with δ _Q = 1 then 3Q is disjoint from E. We denote z ₀ ≔ (−1, 0), w ₀ ≔ (1, 0) to be the points of maximal separation in E ₁. Note that

We say that a square Q ∈ CZ is of Type I if #(1.1Q ∩ E ₁) = 1, Type II if #(1.1Q ∩ E ₂) = 1, and Type III if #(1.1Q ∩ E) = 0. We denote the collection of squares of Type I, II, and III by CZ_I, CZ_II, and CZ_III, respectively. These collections form a partition of CZ because #(3Q ∩ E) ≤ 1 for any Q ∈ CZ by construction.

Observe that ∂CZ ⊂ CZ_III.

From Property 1 of Lemma 3, it is easy to deduce that

From Property 5 of Lemma 3 we have

Combining this with (7), and using that points of E are Δ-separated, we get

Combining this with (7) and (9) gives

For each Q ∈ CZ_II we let x _Q be the unique point in (1.1Q) ∩ E = (1.1Q) ∩ E ₂. Note that x _Q is undefined for Q ∈ CZ\CZ_II.

For each Q ∈ CZ we associate a pair of points z _Q , w _Q ∈ E ₁. We list the key properties of these points in the next lemma.

4 Constructing the interpolant

Let f : E → R . In this section, we will construct a function F : Q 0 → R satisfying F| _E = f.

Let { θ Q } Q ∈ CZ be a partition of unity subordinate to CZ constructed so that the following properties hold. For any Q ∈ CZ, we have:

• (POU1) supp(θ _Q ) ⊂ 1.1Q.

• (POU2) For any |α| ≤ 2, ‖ ∂ α θ Q ‖ L ∞ ≲ δ Q − | α | .

• (POU3) 0 ≤ θ _Q ≤ 1.

  For any x ∈ Q ⁰, we have

• (POU4) ∑ _Q∈CZ θ _Q (x) = 1.

Our interpolant F will then be of the form

where each P _Q is an affine polynomial on R 2 of the form

Here, L _Q is an affine polynomial satisfying ∂₂ L _Q = 0 and η _Q is a real number.

In the previous section we have associated to each square Q ∈ CZ the points z _Q , w _Q ∈ E ₁. We now define L _Q to be the unique affine polynomial satisfying:

Thanks to Property 4 of Lemma 4 there exists an affine polynomial L ₀ for which

We will choose the η _Q later in Sections 6 and 7.

We now compute a simple upper bound for the L ^2,p -seminorm of F.

5 Establishing compatibility of the L _Q

In this section we will prove that for any function G ∈ L 2 , p ( R 2 ) with G| _E = f we have

Fix some r satisfying 1 < r < p. We claim that for any Q, Q′ ∈ CZ with Q ↔ Q′ we have

Observe that for any x ∈ Q we have

Since G z Q ′ = f ( z Q ′ ) , and since z Q ( 2 ) = z Q ′ ( 2 ) = 0 , we have

Therefore, since ∂₂ L _Q = ∂₂ L _Q′ = 0, we have

Recall that for any Q, Q′ ∈ CZ with Q ↔ Q′ we have 1 2 δ Q ′ ≤ δ Q ≤ 2 δ Q ′ (see Lemma 3), and therefore 50Q′ ⊂ 250Q. In particular, we have

Since G| _E = f, Lemma 2 then implies

we deduce (14).

The inequality (14) implies that

where M is the Hardy–Littlewood maximal operator (see Section 2). Taking (p/r)th powers and integrating gives

By Properties 2 and 3 of Lemma 3, we deduce

Since CZ is a pairwise disjoint collection of squares, we have

We use the boundedness of the Hardy–Littlewood maximal operator from L ^q to L ^q for 1 < q ≤ ∞ to deduce (13).

6 Step I of choosing the η _Q : clustering

For each 0 ≤ ℓ ≤ L and fixed s ⃗ ≤ ℓ = ( s 1 , … , s ℓ ) ∈ { − 1 , + 1 } ℓ we define a set C ( s ⃗ ≤ ℓ ) ≔ ( z , Δ ) : z = ∑ k = 1 L s k ε k , s ℓ + 1 , … , s L ∈ { − 1 , + 1 } ⊂ E 2 . We refer to C ( s ⃗ ≤ ℓ ) as a cluster of E ₂ at depth ℓ. (By convention, in the edge case ℓ = 0, s ⃗ ≤ 0 is the empty list and C ( s ⃗ ≤ 0 ) = E 2 .) Note that the clusters of depth L are singletons.

Accordingly, for fixed ℓ there are 2 ^ℓ distinct clusters of E ₂ of depth ℓ. The convex hulls of distinct clusters are disjoint (assuming ɛ ≤ 1/2), and thus, the clusters of depth ℓ inherit an order from the real number line. We shall order the clusters of E ₂ of depth ℓ by C 1 ℓ , C 2 ℓ , …, C 2 ℓ ℓ – the following properties are apparent:

1. # C i ℓ = 2 L − ℓ ,

2. d i a m C i ℓ ≈ ε ℓ + 1 (for ℓ = 0, …, L − 1), and

3. C i ℓ i = 1 2 ℓ is a partition of E ₂.

We write C ℓ ≔ C i ℓ i = 1 2 ℓ to denote the set of all clusters of E ₂ of depth ℓ. We then define the set of all clusters

The set of all clusters forms a full, binary tree via the relation of set inclusion, with clusters of depth ℓ corresponding to vertices of depth ℓ. Any C ∈ C ℓ with ℓ ≥ 1 therefore has a unique parent cluster π ( C ) ∈ C ℓ − 1 .

To every cluster C ∈ C ℓ for ℓ = 0, …, L − 1 we associate a ball B _C and to every cluster C ∈ C ℓ for ℓ = 1, …, L − 1 we associate a ball B ̂ C . For convenience, we’ll assume that B _C , B ̂ C are centered at the same point and that radius ( B ̂ C ) = 1 0 M + 1 ⋅ radius ( B C ) for a positive integer M (independent of the cluster C). We require these balls to satisfy the following properties:

1. For every C ∈ C ℓ (ℓ = 0, …, L − 1), we have

   1. diam(B _C ) ≈ diam(C) ≈ ɛ ^ℓ+1, and

   2. C ⊂ B _C .

2. For every C ∈ C ℓ (ℓ = 1, …, L − 1), we have

   1. d i a m ( B ̂ C ) ≈ d i a m ( π ( C ) ) ≈ ε ℓ , and

   2. B C ⊂ B ̂ C ⊂ B π ( C ) .

3. For distinct clusters C , C ′ ∈ C ℓ , we have

   1. dist(B _C , B _C′) > cɛ ^ℓ (ℓ = 0, …, L − 1), and

   2. dist ( B ̂ C , B ̂ C ′ ) > c ε ℓ (ℓ = 1, …, L − 1).

We remark that (after taking ɛ smaller than some absolute constant) the last bullet point implies that, for fixed ℓ, each of the families { B C } C ∈ C ℓ and { B ̂ C } C ∈ C ℓ is pairwise disjoint.

Given f : E → R , for each x = (x ⁽¹⁾, Δ) ∈ E ₂ we define

For each Q ∈ CZ_II there is a single point in (1.1Q ∩ E ₂), denoted by x _Q . We then define

To any square Q ∈ CZ we associate a cluster C _Q as follows. If Q ∈ CZ\CZ_II, then we define C _Q to be equal to the cluster C ∈ C ℓ (ℓ = 0, 1, …, L − 1) of smallest diameter for which Q ⊂ B _C if such a cluster exists and equal to E ₂ (recall that E ₂ is the unique cluster of depth 0) if no such cluster exists. If Q ∈ CZ_II, then we set C Q = { x Q } ∈ C L , where x _Q is the unique point in 1.1Q ∩ E ₂. Observe that for all Q ∈ ∂CZ we have C _Q = E ₂.

Observe that C Q ∈ C L if and only if 1.1Q ∩ E ₂ ≠ ∅. In particular, if 1.1Q ∩ E ₂ = ∅ then C Q ∈ C ℓ (ℓ = 0, …, L − 1) has cardinality at least 2.

For each cluster C ∈ C we introduce a real number η _C . Recall that every cluster C ∈ C L satisfies #(C) = 1; we let x _C denote the unique point in C. We then define

(Recall that η _x is defined by (15).) We defer choosing { η C } C ∈ C ℓ for ℓ = 0, …, L − 1 until Section 7.