Linear extension operators for Sobolev spaces on radially symmetric binary trees

1 Introduction

Consider a full binary tree of height N , whose set of vertices is denoted by

i.e., the vertices are strings of zeroes and ones of length at most N . For x ∈ { 0 , 1 } k and ℓ ≤ k , we write π ℓ ( x ) ∈ { 0 , 1 } ℓ for the prefix of x of length ℓ . Thus, for k ≥ 1 , the parent of a vertex x ∈ { 0 , 1 } k ⊆ V is the vertex π k − 1 ( x ) . The set { 0 , 1 } 0 is a singleton whose unique element is denoted by ∅ , the empty string, which is the root of the tree. The set of leaves of the tree is { 0 , 1 } N , and all other vertices in V are internal vertices. A vertex x ∈ { 0 , 1 } k is said to have depth

thus leaves have depth N and the root has depth 0. The collections of bijections from V to V that preserve depth and parenthood relations form a group. This group is referred to as the symmetry group of the tree. It has 2 2 N − 1 elements, and it is a 2-Sylow subgroup of the group of all permutations of the leaves. The set

is referred to as the set of edges of the tree, and the depth of an edge e ∈ { 0 , 1 } k is d ( e ) = k . That is, we think of e ∈ E as an undirected edge connecting the vertex whose string corresponds to e to its unique parent. Each internal vertex other than the root is connected to three vertices, which are its parent and its two children. Assume that we are given edge weights

where we view W k as the weight of all edges of depth k . For 1 < p < ∞ , the associated W ˙ 1 , p -seminorm is defined, for F : V → R , via

We write ∂ V = { 0 , 1 } N ⊆ V for the set of leaves of the tree. The trace of the ‖ ⋅ ‖ W ˙ 1 , p -seminorm is defined, for f : ∂ V → R , via

i.e., the infimum of the W ˙ 1 , p -seminorm over all extensions of f from the leaves to the entire tree. We write R V for the collection of all functions f : V → R , and similarly R ∂ V is the collection of all functions f : ∂ V → R . Our main result is the following:

The proof of Theorem 1.2 is constructive, and the extension operator H that we construct is in fact a harmonic extension operator with respect to a certain random walk defined on the tree. At each step, the random walk jumps from a vertex to one of its neighbors, where of course the neighbors of a vertex are its parent and its children. The Markov kernel corresponding to the random walk is invariant under the symmetries of the tree; thus, the probability to move from a vertex to its neighbor depends only on the weights of the vertex and of its neighbor. The Markov kernel of our random walk is determined by the following requirement: For any s = 1 , … , N , the probability that a random walk starting at some vertex of depth s will reach a leaf before reaching a vertex of depth s − 1 equals

Thus, the weights of our random walk typically depend on p ∈ ( 1 , ∞ ) , except for the case where W s is proportional to 2 − s . This seems inevitable. Indeed, in some examples such as the 3rd example in Section 2, the linear extension operator H : R ∂ V → R V that corresponds to the parameter value p = p 0 is not uniformly bounded for any p ∈ ( 1 , ∞ ) \ { p 0 } . When p = 2 , our random walk coincides with the usual random walk corresponding to the given weights on the edges of the binary tree, and thus in this case, H is the standard harmonic extension operator (hence, C ¯ p = 1 for p = 2 ).

There is a certain range of weights where the averaging operator yields a uniformly bounded linear extension operator, as proven by Björn et al. [1]. The averaging operator is the extension operator that assigns to each internal vertex v the average of the function values on the leaves of the subtree whose root is v . This averaging operator seems natural also from the point of view of Whitney’s extension theory, and see the work by Shvartsman [5] on Sobolev extension in W 1 , p ( R n ) . However, there are examples of radially-symmetric binary trees where the averaging operator does not provide a uniformly bounded operator, such as the case where W k = 2 − k for all k .

What about trees with weights that are not radially symmetric? Suppose that the edge weights are arbitrary positive numbers ( W e ) e ∈ E that are not necessarily determined by the depth of the edge. When p = 2 , there is still a harmonic extension operator of norm one from W ˙ 1 , p ( ∂ V ) to W ˙ 1 , p ( V ) . However, for p ≠ 2 , the situation seems subtle. We conjecture that in the general case of nonradially-symmetric tree weights, there is no linear extension operator whose norm is bounded by a function of p alone. This conjecture is closely related to questions about well-complemented subspaces of ℓ p n that are beyond the scope of this note.

To prove Theorem 1.1, we reformulate the problem in a way that brings us closer to analysis in ℓ p -spaces. For 1 < p < ∞ , the associated L p ( E ) -norm is defined, for f : E → R , via

We think of a function f : E → R as the gradient of a function f ˜ : V → R , uniquely determined up to an additive constant. Given f : E → R , we thus define a function f ˜ : V → R as follows: For any x ∈ V other than the root,

while f ˜ ( x ) = 0 if x = ∅ is the root. The only property of f ˜ that matters is that for all x ∈ E ,

We are interested in finding a linear operator T : L p ( E ) → L p ( E ) , with a uniform bound on its operator norm, which has the following properties:

1. The operator T takes the form

   (7) T f ( x ) = T ˜ f ˜ ( x ) − T ˜ f ˜ ( π d ( x ) − 1 x )

   for some linear operator T ˜ : R V → R V . That is, T ˜ takes functions on V to functions on V , and T is induced from T ˜ via formula (7).

2. The operator T ˜ is equivariant with respect to the tree symmetries and it satisfies T ˜ ( 1 ) ≡ 1 , i.e., it maps the constant function 1 to itself.

3. The function T ˜ g coincides with the function g on the leaves of the tree, i.e., ( T ˜ g ) ∣ ∂ V = g ∣ ∂ V for any g : V → R .

4. The function T ˜ g is determined by the values of the function g on the leaves of the tree.

The operator norm of T ˜ with respect to the W ˙ 1 , p -seminorm equals to the operator norm of T with respect to the L p ( E ) -norm. Defining H ( f ∣ ∂ V ) = T ˜ f , Theorem 1.1 may thus be reformulated as follows:

The proof of Theorem 1.2 occupies the next three sections. In Section 2, we discuss invariant random walks on a full binary tree and describe the corresponding harmonic extension operator. In Section 3, we deal with the problem of bounding the norm of this operator and use the symmetries of the problem to reduce it to a one-dimensional question. This one-dimensional question is then answered in Section 4 using the Muckenhoupt criterion [4].

When analyzing the binary tree, we use the following notation: We write dlca ( x , y ) for the length of the maximal prefix shared by the strings x ∈ { 0 , 1 } k and y ∈ { 0 , 1 } ℓ , while l c a ( x , y ) is the maximal prefix itself. Thus for two vertices x , y ∈ V , their least common ancestor is l c a ( x , y ) ∈ V and its depth is dlca ( x , y ) ∈ { 0 , … , N } . Note that for any x ∈ E and ω ∈ ∂ V ,

2 Invariant random walks

A Markov chain on V is a sequence of random variables R 1 , R 2 , … ∈ V such that the distribution of R i + 1 conditioned on R 1 , … , R i is the same as its distribution conditioned on R i . A Markov chain is time-homogeneous if for any x , y ∈ V , the probability that R i + 1 = x conditioned on the event R i = y does not depend on i . A random walk on V is a time-homogeneous Markov chain R 1 , R 2 , … ∈ V such that R i + 1 is a neighbor of R i with probability one.

We say that the random walk is invariant if the probability to jump from a vertex x to a vertex y depends only on the depths d ( x ) and d ( y ) . Our random walk will be invariant, and it will stop when it reaches a leaf, i.e., we have the stopping time

For s ≥ 1 , we define q s to be the probability of the following event: Assuming that R 1 is a vertex of depth s , the event is that R i will remain at the subtree whose root is R 1 for all 1 ≤ i ≤ τ . Equivalently, define

Then X 1 , X 2 , … ∈ { 0 , … , N } is a random walk, since ∣ X i + 1 − X i ∣ = 1 for all i . Furthermore,

Clearly,

For r , s ∈ { 0 , … , N } with r ≤ s , we set

That is, the number p s , r is the probability that r is the minimal node that the walker visits when starting from node s , before reaching the terminal node N . Clearly, ∑ r = 0 s p s , r = 1 .

Suppose that our random walk R 1 , R 2 , … ∈ V begins at a vertex R 1 = x with d ( x ) = s . Consider a leaf y ∈ ∂ V with dlca ( x , y ) = r . What is the probability that our random walk will reach the leaf y ? We claim that this probability is

Indeed, conditioning on the value of k = min i ≤ τ d ( R i ) , by symmetry, we know that R τ is distributed uniformly among the 2 N − k leaf descendants of the vertex π k ( x ) . When k ≤ r , exactly one of these leaf-descendants is the leaf y , since the vertex of minimal depth that ( R i ) visits must be the vertex π k ( x ) , which is a prefix of y as k ≤ r = dlca ( x , y ) . Hence, the probability that R τ = y , conditioning on the value of k , equals to 1 / 2 N − k when k ≤ r , and it vanishes otherwise. By using the definition of p s , k and the complete probability formula, we obtain (13). The harmonic extension operator associated with our invariant random walk is given by

The operator T is induced from T ˜ via formula (7). Requirements 1 , … , 4 from Section 1 are clearly satisfied.

We stipulate that the collection of descendants of a vertex x ∈ V , denoted by D ( x ) ⊆ V , includes the vertex x itself. Abbreviate a ∧ b = min { a , b } and a ∨ b = max { a , b } . The operator T takes the form

where the kernel K is described next.

Some examples.

1. The simplest example is when q s = 1 for all s ≥ 1 . In this case, the operator T ˜ is the familiar averaging operator. That is, the extension operator T ˜ is the operator that assigns to each vertex the average of the values at the leaves of its subtree. In this case,

   p s , r = δ s , r .

2. Consider the case where the invariant random walk is such that X i ≔ d ( R i ) is a symmetric random walk on { 0 , … , N } , i.e., the probability to jump from i to i + 1 is exactly 1/2 for i = 1 , … , N − 1 . Recall that q s is the probability to never leave the subtree when starting at a vertex of depth s . We claim that in this example, for s = 1 , … , N ,

   (23) q s = 1 N − s + 1 .

   Indeed, the function f ( i ) = i is harmonic on { 0 , 1 , … , N } , and hence, f ( X i ) is a martingale. Thus, for any stopping time τ ˜ , we have f ( X 1 ) = E f ( X τ ˜ ) . We pick the stopping time

   τ ˜ = min { i ; X i ∈ { s − 1 , N } }

   and obtain (23) since

   N ⋅ q s + ( s − 1 ) ⋅ ( 1 − q s ) = s .

   Next, we use Lemma 2.1 and find a formula for p s , r . Since formula (23) is valid for any s ≥ 1 , we conclude that for any r ≥ 0 and s ≥ r + 1 ,

   (24) ∏ k = r + 1 s ( 1 − q k ) = ∏ k = r + 1 s N − k N − k + 1 = N − s N − r .

   Formula (24) is actually valid for any 0 ≤ r ≤ s , since an empty product equals one. Recall that q 0 = 1 . We thus conclude from Lemma 2.1 that for s = 0 , … , N − 1 ,

   p s , r = N − s N − r ⋅ 1 r = 0 1 N − r + 1 1 ≤ r ≤ s ,

   while p N , r = δ N , r .

3. Let 0 < δ < 1 , and consider the case where ( X i ) is a random walk on { 0 , … , N } such that the probability to jump from k to k + 1 equals 1/2 if k < N − 1 , and it equals δ if k = N − 1 . A harmonic function here is

   f ( k ) = k k ≤ N − 1 N − 2 + 1 / δ k = N .

   Therefore, for s = 1 , … , N − 1 ,

   ( N − 2 + 1 / δ ) ⋅ q s + ( s − 1 ) ( 1 − q s ) = s .

   Thus, q 0 = q N = 1 , while for s = 1 , … , N − 1 ,

   q s = 1 N − s + 1 / δ − 1 .

   Hence, for any s ≤ N − 1 and r ≤ s − 1 ,

   ∏ k = r + 1 s ( 1 − q k ) = ∏ k = r + 1 s N − k + δ − 1 − 2 N − k + δ − 1 − 1 = N − s + δ − 1 − 2 N − r + δ − 1 − 2 .

   We conclude from Lemma 2.1 that for s = 0 , … , N − 1 and 0 ≤ r ≤ s ,

   p s , r = q r ⋅ ∏ k = r + 1 s ( 1 − q k ) = N − s + δ − 1 − 2 N − r + δ − 1 − 2 ⋅ 1 r = 0 1 N − r + 1 / δ − 1 1 ≤ r ≤ s ,

   while p N , r = δ N , r .

We conclude this section with the following:

3 The ancestral and nonancestral parts of the kernel

We need to bound the operator norm in L p ( E ) of the operator T whose kernel is described in Proposition 2.2. Let us consider first the nonancestral part of the operator, given by the kernel

Here, as usual s = d ( x ) , t = d ( y ) , and r = dlca ( x , y ) . Write T 0 : L p ( E ) → L p ( E ) for the operator whose kernel is K 0 . A function f : E → R is invariant under the symmetries of the tree, or invariant in short, if it takes the form

for some function F : { 1 , … , N } → R . The operator T 0 is equivariant under the symmetries of the tree. Therefore, if f ( x ) = F ( d ( x ) ) is an invariant function, then so is T 0 f . In fact, in the case where f ( x ) = F ( d ( x ) ) , we can write

for a certain kernel L 0 ( s , t ) defined for s , t = 1 , … , N .