Sobolev extension in a simple case

1 Introduction

Continuing from [1], [2], [3], [4], [5], [6], we study the Sobolev extension problem: given a finite set E ⊂ R n and a function f : E → R , our task is to find a function F in the homogeneous Sobolev space L m , p ( R n ) satisfying F = f on E such that ‖ F ‖ L m , p ( R n ) has the least possible order of magnitude. Moreover, we want to compute the order of magnitude of ‖ F ‖ L m , p ( R n ) . Here, L m , p ( R n ) denotes the homogeneous Sobolev spaces consisting of functions F on R n whose distributional derivatives of order m belong to L ^p , with seminorm

Notice that when p > n m , any function F ∈ L m , p ( R n ) is continuous, so the restriction F| _E to any finite subset E ⊂ R n is well-defined. Thus, our requirement that F = f on E makes sense for any choice of f and our problem is well-posed provided that p > n m .

In the narrower parameter range p > n, the problem is well-understood; see Israel [3], Fefferman-Israel-Luli [1], [2] and Drake [7]. We recall the relevant results as follows. Let L ^m,p (E) denote the space of functions f : E → R equipped with the seminorm

For p > n and E ⊂ R n finite, there exist a linear map T : L m , p ( E ) → L m , p ( R n ) , and linear functionals ℓ 1 , … , ℓ ν max : L m , p ( E ) → R , with the following properties:

1. T f = f    on  E  and  ‖ T f ‖ L m , p ( R n ) ≤ C m , n , p ‖ f ‖ L m , p ( E )    for all  f ∈ L m , p ( E ) .

2. Let f p ≔ ∑ ν = 1 ν max | ℓ ν ( f ) | p 1 / p . Then

   c m , n , p f p ≤ ‖ f ‖ L m , p ( E ) ≤ C m , n , p f p    for all  f ∈ L m , p ( E ) .

   Here, the constants C _m,n,p and c _m,n,p depend only on m, n, p, the parameters specifying the Sobolev space.

3. Moreover, the number ν _max of linear functionals ℓ _ν grows only linearly with N, the number of points of E, and the ℓ _ν have a sparse structure that lets us evaluate f p in O(N) computer operations. The functionals ℓ _ν and the operator T can be computed efficiently. See [1] for details.

When n m < p ≤ n , the proofs of the above results break down, and we know almost nothing. In this paper, we analyze the simplest non-trivial case. We work in the homogeneous Sobolev space L 2 , p ( R 2 ) for 1 < p < 2. We suppose our finite set E is contained in a line. Let (x, y) denote the canonical orthogonal coordinates in R 2 ; by rotating our coordinates, we may assume that E lies on the x-axis. Our main results are as follows.

Fix p (1 < p < 2). Let E ⊂ R × { 0 } ⊂ R 2 be finite, with N = #E ≥ 2. We write c, C, C′, etc. to denote absolute constants, and we write c p , C p , C p ′ , etc. to denote constants depending only on p.

Further, under the same assumptions as in Theorem 1, we can efficiently compute the order of magnitude of ‖ f ‖ L 2 , p ( E ) :

The remainder of the paper is dedicated to the proof of Theorem 1 and Theorem 2.

Before proceeding, let us first outline an important issue arising in the proof of Theorem 1 and Theorem 2 and the corresponding key new ingredient introduced here to deal with this. Note that this discussion is merely an overview; see the relevant succeeding sections for a more precise description. Fix 1 < p < 2 and suppose we are given f : E → R . We will make a Calderón–Zygmund (henceforth abbreviated by CZ) decomposition in the plane. Figure 1 depicts two possible scenarios for a CZ square Q, whose distance to the x-axis is comparable to its side length δ _Q , together with nearby points in E. In Figure 1a, |z ₁ − z ₂| is comparable to δ _Q , while in Figure 1b, |z ₁ − z ₂| is much smaller than δ _Q , while |z* − z ₁| is much larger than δ _Q .

Figure 1:

Two possible scenarios for a CZ square Q. (a) z ₁, z ₂ ∈ E near Q with |z ₁ − z ₂|≃ δ _Q . (b) z ₁, z ₂, z* ∈ E near Q with |z ₁ − z ₂|≪ δ _Q and |z* − z ₁|≫ δ _Q .

More precisely, the points z ₁, z ₂, z* belong to E and all other points of E lie much farther away from Q than those depicted. Let F ∈ L 2 , p ( R 2 ) satisfy F = f on E with ‖ F ‖ L 2 , p ( R 2 ) being of least possible order of magnitude. What can we infer about the behavior of F on Q? In particular, what can we say about ∂ _x F on Q?

If z ₁, z ₂ and Q are as in Figure 1a, one expects that ∂ _x F| _Q is closely approximated by the difference quotient

The comparability of |z ₁ − z ₂| to δ _Q combined with a Sobolev inequality allows one to control ‖ ∂ x F − DQ ( z 1 , z 2 ) ‖ L p ( Q ) by the product δ Q ⋅ ‖ ∇ 2 F ‖ L p ( C Q ) , where CQ is the dilation of Q about its center by an appropriate constant factor C > 1 such that CQ contains z ₁, z ₂.

On the other hand, suppose that z ₁, z ₂, z* and Q are as in Figure 1b. This time, since |z ₁ − z ₂|≪ δ _Q , DQ(z ₁, z ₂) is not an effective approximation for ∂ _x F| _Q . Instead, the difference quotient DQ z 1 , z * is a better choice of approximant for ∂ _x F| _Q . We may again control ‖ ∂ x F − DQ z 1 , z * ‖ L p ( Q ) by the product δ Q ⋅ ‖ ∇ 2 F ‖ L p ( Q * ) for a square Q* containing z ₁, z*, but this time Q* is much larger than Q, since |z* − z ₁|≫ δ _Q . Therefore, in order to avoid having unbounded overlaps when summing over all squares Q, we cannot simply consider a single such square Q associated to Q*.

Instead, for a fixed z* ∈ E, we group together all CZ squares Q for which a scenario like that in Figure 1b holds for z*. We denote this subcollection of the CZ squares by GP(z*). Thanks to a lemma in Section 8, the structure of the squares in GP(z*) looks like that depicted in Figure 2.

Figure 2:

CZ squares Q ₁, Q ₂, Q ₃ ∈ GP(z*) for a point z* ∈ E.

This allows us to estimate

by ‖ ∇ 2 F ‖ L p ( Q * ) p for a fixed z* ∈ E. This is a fundamental difference between the case 1 < p < 2 in consideration here, and the aforementioned case p > 2. Namely, when p > 2 we can make use of the difference quotient DQ(z ₁, z ₂) even in the setting of Figure 1b, without taking into account z*. This allows us to study each CZ square Q in isolation, then sum over all Q together.

So our main new ingredient is the introduction of GP(z*). By making a translation and dilation of E, we may assume without loss of generality that E = E ̲ × { 0 } ⊂ R 2 for a finite subset E ̲ with max E ̲ = 2 − 11 and min E ̲ = − 2 − 11 . We then work with the set E ⁺ = E ∪ {(−1, 0)}, so that we may guarantee the existence of a point z* ∈ E as in Figure 1b to the left of any CZ square Q. At the end of the proof in Section 11, we remove this additional point (−1, 0).

2 Main result

We will prove the following theorem, which is equivalent to Theorems 1 and 2.

3 Constants

Let 1 < p < 2. Fix p ̃ = 1 + p 2 ∈ ( 1 , p ) . We write c, C, C′ to denote absolute constants and we write c p , C p , C p ′ to denote constants depending only on p. We may use the same symbol to denote different constants in different occurrences.

Let K > 0 be larger than some absolute constant. We write

to denote constants depending only on K. Once again, the same symbol could be used to denote different occurrences.

At the end of Section 8, we choose K to be an absolute constant, taken sufficiently large. Once we do so, K and any instances of C(K) become absolute constants C.

4 Dyadic squares and intervals

Here, dyadic squares are Q 0 = [ − 1,1 ] 2 ⊂ R 2 and all closed squares arising from Q ⁰ by successive bisection. More precisely, any dyadic square Q is either Q ⁰ or of the form [i2^−k , (i + 1)2^−k ] × [j2^−k , (j + 1)2^−k ], for some k ∈ N ∪ { 0 } and i, j ∈ {−2 ^k , …, 2 ^k − 1}. Similarly, dyadic intervals consist of the interval I 0 = [ − 1,1 ] ⊂ R and all the closed intervals [i2^−k , (i + 1)2^−k ] for i ∈ {−2 ^k , …, 2 ^k − 1} and k ∈ N , obtained by successive bisection of I ⁰. Note that dyadic squares and intervals are closed.

If Q is any dyadic square, we denote its side length by δ _Q and if I is a dyadic interval, we denote its length by |I|. Moreover, we denote the area δ Q 2 of a dyadic square Q by |Q|. Each dyadic square Q of side length δ _Q ≤ 1 arises by bisecting a dyadic square Q ⁺ ⊃ Q of side length 2δ _Q ; we call Q ⁺ the dyadic parent of Q. Similarly, each dyadic interval I of length |I| ≤ 1 arises by bisecting a dyadic interval I ⁺ ⊃ I of length 2|I|; we call I ⁺ the dyadic parent of I.

If I is a dyadic interval and C > 1, then CI denotes the open interval of length C|I| whose center coincides with that of I. Similarly, if Q is a dyadic square and C > 1/4 (C ≠ 1), then CQ denotes the open square of side length Cδ _Q whose center coincides with that of Q.

Throughout this article, we will make frequent use of the square Q inner ≔ [ − 2 − 10 , 2 − 10 ] 2 ⊂ Q 0 .

5 Some elementary inequalities

Suppose F ∈ L 2 , p ( R 2 ) . In this section, we let Q denote a closed square in R 2 . We will henceforth simply refer to such Q as a square. For any such Q ⊂ R 2 , we have the Sobolev Inequality [9], Lemma 7.16]: F may be expressed in the form F ₁ + L on Q, where L is a first degree polynomial and F ₁ is continuous on Q; moreover,

For instance, given any point z ₀ ∈ Q, one may take L ( z ) = F ( z 0 ) + 1 | Q | ∫ w ∈ Q ∇ F ( w ) d w ⋅ ( z − z 0 ) . This inequality is not sharp, but it’s enough for our purposes.

We introduce the (non-centered) maximal function

for functions φ : R 2 → R . Here the supremum is taken over all squares Q containing z.

We recall the Hardy–Littlewood Maximal Theorem [10], Chapter I, §3, Theorem 1]:

For F ∈ L 2 , p ( R 2 ) and a square Q ⊂ R 2 , we write Av _Q ∇F to denote the mean 1 | Q | ∫ Q ∇ F ( z ) d z . Similarly, we write Av _Q ∂ _x F and Av _Q ∂ _y F to denote the components of Av _Q ∇F.

Applying the Sobolev Inequality to the open square CQ (C > 1.1) defined as in Section 4, we obtain the following corollaries:

We use also the following estimates:

The elementary proof is left to the reader.

We leave the proof of this inequality to the reader.

6 The set E

Let E = E ̲ × { 0 } ⊂ R 2 , where E ̲ ⊂ R is a finite set. Let N = #E ≥ 2. We suppose max E ̲ = 2 − 11 and min E ̲ = − 2 − 11 .

We will find it convenient to add the point (−1, 0) to E; we set E ̲ + = E ̲ ∪ { − 1 } , and let E + = E ̲ + × { 0 } .

We will work with functions f : E → R and with functions f + : E + → R .

7 The Calderón–Zygmund decomposition

Starting with the square Q ⁰, we perform repeated bisection à la Calderón–Zygmund, stopping at a square Q provided #(E ∩ 3Q) ≤ 1. Recall 3Q denotes an open square of side length 3δ _Q . This process results in a decomposition of Q ⁰ consisting of finitely many closed Calderón–Zygmund squares. We write CZ to denote the set of Calderón–Zygmund squares. Equivalently, the CZ squares are precisely the maximal dyadic squares Q such that #(E ∩ 3Q) ≤ 1.

We write Q ↔ Q′ for CZ squares Q, Q′ to indicate that Q ∩ Q′ ≠ ∅. If the CZ squares Q, Q′ satisfy Q ∩ Q′ ≠ ∅, then 1 2 δ Q ≤ δ Q ′ ≤ 2 δ Q . We recall the standard proof: Suppose Q ∩ Q′ ≠ ∅ with δ Q ′ ≤ 1 4 δ Q . Then the dyadic parent (Q′)⁺ of Q′ satisfies 3(Q′)⁺ ⊂ 3Q. Since #(E ∩ 3Q) ≤ 1, it follows that #(E ∩ 3(Q′)⁺) ≤ 1, and consequently we should not have bisected (Q′)⁺ to arrive at the CZ square Q′. This contradiction shows that δ Q ′ ≥ 1 2 δ Q . By the same reasoning, δ Q ≥ 1 2 δ Q ′ . In light of this “good geometry” we recall a familiar bound:

Patching Estimate [1], Lemma 9.1], [8], Lemma 5]: Let ( θ Q ) Q ∈ C Z ⊂ C ∞ ( Q 0 ) be a smooth partition of unity satisfying for Q ∈ CZ, 0 ≤ θ _Q ≤ 1; θ _Q vanishes on Q ⁰ \ 1.1Q; | ∂ α θ Q | ≤ C δ Q − | α | for |α| ≤ 2; θ Q | 1 2 Q ≡ 1 | 1 2 Q ; and ∑ _Q∈CZ θ _Q ≡ 1 on Q _inner. Suppose G _Q ∈ L ^2,p (1.1Q) for each Q ∈ CZ satisfying 1.1Q ∩ Q _inner ≠ ∅. We define the function G : Q inner → R as G(z) : = ∑ _Q∈CZ θ _Q (z)G _Q (z). We have

Let π : R 2 → R be the projection (x, y) ↦ x. The shadow of a dyadic square Q is the dyadic interval π(Q).

We make the following observation regarding shadows:

Figure 3:

Two possible scenarios for a contactless child Q′ of Q ⁺⁺.

The square Q ⁰ satisfies #(E ∩ 3Q ⁰) ≥ 2 since (−2⁻¹¹, 0), (2⁻¹¹, 0) ∈ E. Hence every CZ square Q arises by bisecting Q ⁺, so #(E ∩ 3Q ⁺) ≥ 2, else we wouldn’t have bisected Q ⁺. This means that # ( E ̲ ∩ 3 I + ) ≥ 2 , where I ⁺ = π(Q ⁺).

For each shadow I, we fix a point x ̂ ( I ) ∈ E ̲ ∩ 3 I + ⊂ E ̲ ∩ 7 I .

Let Q be a CZ square. We say that Q is relevant if 1.1Q ∩ Q _inner ≠ ∅, see Section 4, and we denote the set of relevant CZ squares by CZ _rel

We note that every relevant CZ square Q satisfies δ _Q ≤ 2⁻⁸. Indeed if δ _Q ≥ 2⁻⁷ and 1.1Q ∩ Q _inner ≠ ∅, then 3Q ⊃ Q _inner. Hence #(E ∩ 3Q) ≥ 2, so Q cannot be a CZ square.

A relevant shadow is defined to be the shadow of a relevant CZ square. relevant shadows I thus satisfy 1.1I ∩ [−2⁻¹⁰, 2⁻¹⁰] ≠ ∅ and |I| ≤ 2⁻⁸.

8 Groups of shadows and squares

Let I = π(Q) be the shadow of Q ∈ CZ.

Recall, from Section 3 we assume K > 0 larger than a universal constant. We say that I is easy if

Otherwise, we say that I is hard.

Fix x * ∈ E ̲ + . Then we define Gp(x*) (the “group” of x*) to be the set of all shadows I such that

1. x* ≤ inf  41I, and

2. E ̲ + ∩ ( x * , inf 9 I ] = ∅ .

The following elementary lemma will be useful when decomposing shadows into disjoint subfamilies.

Until further notice, we fix a point x * ∈ E ̲ + and study its group Gp(x*). We assume that Gp(x*) ≠ ∅.

Fix a shadow I ₀ ∈ Gp(x*) with

Now consider the family I ₀ ⊂ I ₁ ⊂ … ⊂ I _L , where each I _ℓ (ℓ ≥ 1) is the dyadic parent of I _ℓ−1, and L is the largest integer for which all the I _ℓ (ℓ ≤ L) belong to Gp(x*). We first demonstrate that I _L satisfies the hypothesis of Lemma 8.3 with c = 1 43 .

We will henceforth fix c = 1 43 in Lemma 8.3. Then, by Lemma 8.3, if J ∈ Gp(x*) is hard, we must have | J | ≤ 1 42 dist ( x * , I 0 ) < | I L | . Consequently, for some ℓ (0 ≤ ℓ ≤ L − 1), we have |J| = |I _ℓ |. Since J and I _ℓ belong to Gp(x*), Lemma 8.2 tells us that 9J ∩ 9I _ℓ ≠ ∅.

We now pass from groups of shadows to groups of CZ squares.

Collecting the above results regarding the properties of Gp(x*), and noting Lemma 7.1, we obtain the following:

The squares Q ₀, Q ₁, …, Q _L and the number L in Lemma 8.5 depend on z*; we will thus sometimes write Q _ℓ (z*) to denote the square Q _ℓ and emphasize the dependence on z*. We write Q _small(z*) to denote Q ₀ and Q _big(z*) to denote Q _L . We let

The CZ squares Q _ℓ (z*) may or may not be hard. We write L(z*) to denote the number L associated to z* in Lemma 8.5.

We conclude this section by estimating the number of easy CZ squares. Recall that N = #E.