ASYMPTOTIC PROPERTIES OF THE VECTOR CARLESON EMBEDDING THEOREM

MICHAEL GOLDBERG

Abstract. The dyadic Carleson embedding operator acting on ℂ^{n}-valued functions has norm at least C log n. Thus the Carlesson Embedding Theorem fails for Hilbert space valued functions.

Let 𝕋 be the unit cirle in ℂ, and {I}_{I∈D} its collection of dyadic arcs. Let w_{I} be nonnegative real numbers indexed by I ∈ D. For integrable functions f on 𝕋, denote by ⟨f⟩_{I} the average |I|^{−1} ∫_{I} f(y)dy. The classical Carleson embedding theorem [1] is equivalent to the following dyadic result:

Theorem 0. If ∑_{I⊂K} w_{I} ≤ |K| for all K ∈ D, then ∑_{I∈D} w_{I}⟨f⟩_{I}^{2} ≤ C∥f∥^{2} for all f ∈ L^{2}(𝕋).

The converse is also true (up to the placement of constants) and is verified by considering functions of the form f = χ_{J}, J ∈ D.

An analogous statement may be made for functions taking values in ℂ^{n} with matrix-valued weights W_{I} ≥ 0 in the sense of quadratic forms. We wish to consider the following n-dimensional embedding theorem:

Proposition. If ∥∑_{I⊂K} W_{I} ∥ ≤ |K| for all K ∈ D, then ∑_{I∈D}(W_{I}⟨f⟩_{I}, ⟨f⟩_{I}) ≤ C_{n}∥f∥^{2} for all f ∈ L^{2}(𝕋; ℂ^{n}).

The space ℂ^{n} here is viewed as a finite-dimensional Hilbert space. One might ask whether a similar result still holds when f takes values in a general Hilbert space ℍ and W_{I} are positive selfadjoint operators. This is answered in the negative by [4], which proves that C_{n} must be bounded from below by c log n. In the current paper we will use the construction in [4] to verify the stronger bound C_{n} ≥ c(log n)^{2}, which is also proved in [5]. A precise statement is as follows:

Theorem 1. There exist a function f ∈ L^{2}(𝕋; ℂ^{n}) and matrix weights W_{I} ≥ 0 such that ∥∑_{I⊂K} W_{I} ∥ ≤ |K| and ∑_{I∈D}(W_{I}⟨f⟩_{I}, ⟨f⟩_{I}) ≥ c(log n)^{2}∥f∥^{2}, where c > 0 is independent of n.

Remarks. The example presented here is due to Nazarov, Treil, and Volberg [4]. It is further shown in [3] and [4] that the best possible C_{n} is bounded above by C(log n)^{2}, making these results sharp up to a constant factor.

Proof of Theorem 1. Let e_{0}, e_{1}, …, e_{n} be the standard basis for ℂ^{n+1}. Define the Rademacher functions r_{j}(e^{2πit}) = (−1)^{⌊2^{j}t⌋}. For a dyadic interval I, |I| ≤ 2^{−j}, r_{j} is seen to be constant along I. Its value throughout the interval will be called r_{j}(I).

Date: May 9, 2000.

Let f(x) = ∑_{j=0}^{n} r_{j}(x)e_{j}. Clearly ∥f∥^{2} = n + 1. The averages of f over dyadic intervals are also easy to compute. When |I| = 2^{−i}, i ≤ n, ⟨f⟩_{I} = ∑_{j=0}^{i} r_{j}(I)e_{j}.

Let W_{I}, |I| ≥ 2^{−n} be the rank-one operator satisfying W_{I}v = |I|(v, ϕ_{I})ϕ_{I}, where ϕ_{I} = ∑_{j=0}^{i} \frac{1}{i+1−j} r_{j}(I)e_{j}. Define ϕ_{I} to be 0 when |I| < 2^{−n}. Already we can estimate the sum

∑_{I∈D}(W_{I}⟨f⟩_{I}, ⟨f⟩_{I}) = ∑_{I∈D} |I|(⟨f⟩_{I}, ϕ_{I})^{2} = ∑_{i=0}^{n}(∑_{j=0}^{i} \frac{1}{i + 1 − j} )^{2} ≥ cn(log n)^{2}

The only task remaining is to show that ∥ ∑_{I⊂K} W_{I}∥ is controlled by |K|. We will prove the estimate ∑_{I⊂K}(W_{I}v, v) = ∑_{I⊂K} |I|(v, ϕ_{I})^{2} ≤ C|K||v|^{2} for all v ∈ ℂ^{n+1}.

For each interval I with |I| = 2^{−i}, split the vector ϕ_{I} into the sum of two parts, ϕ_{I} = ∑_{j=0}^{k} \frac{1}{i+1−j} r_{j}(K)e_{j} +∑_{j=k+1}^{i} \frac{1}{i+1−j} r_{j}(I)e_{j}. Denote the first sum, which depends only on the length of I ⊂ K, by g_{i}. Summing over all I with |I| = 2^{−i}, and exploiting the orthogonality of the Rademacher functions,

∑_{{I⊂K \atop |I|=2^{−i}}} |I|(v, ϕ_{I})^{2} = |K|((v, g_{i})^{2} + ∑_{j=k+1}^{i} \frac{1}{(i + 1 − j)^{2}} |v_{j}|^{2})

Thus ∑_{I⊂K}(W_{I}v, v) = |K|(∑_{i=k}^{n}(v, g_{i})^{2} + ∑_{j=k+1}^{n} |v_{j}|^{2} ∑_{i=j}^{n} \frac{1}{(i + 1 − j)^{2}} ).

The second sum is less than C|K| ∑_{j=0}^{n} |v_{j}|^{2} = C|K| |v|^{2}. To estimate the first sum, let G represent the (n − k + 1) × (k + 1) matrix whose ij^{th} entry is the coefficient of e_{j−1} in g_{i+k−1}. Then ∑_{i=k}^{n}(v, g_{i})^{2} ≤ ∥G∥^{2}|v|^{2}. Here ∥G∥ is taken as an operator from ℂ^{k+1} to ℂ^{n−k+1}. Under a suitable permutation of indices, however, G is seen to be a restriction of the Hilbert matrix A, (A_{ij} = \frac{1}{i+j−1})|_{i,j=1}^{∞} to finite-dimensional subspaces. It is well known [2] that A is bounded from ℓ^{2}(ℕ) to itself. Thus the first sum is less than |K| ∥A∥^{2}|v|^{2} = C|K| |v|^{2}. Dividing all weights W_{I} by an appropriate constant proves the theorem.

References

[1] J .B. Garnett, Bounded Analytic Functions, Acad. Press, New York, 1981

[2] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, University Press, Cambridge, 1934.

[3] Nets Hawk Katz. Matrix valued paraproducts, J. Fourier Anal. Appl. 300 (1997), pp. 913-921.

[4] F. Nazarov, S. Treil, and A. Volberg, Counterexample to the infinite dimensional Carleson embedding theorem, C.R. Acad. Sci. Paris Sér. I Math., 325 (1997), No.4, pp.383-388.

[5] F. Nazarov, G. Pisier, S. Treil, and A. Volberg, Sharp estimates in the vector Carleson imbedding theorem and for vector paraproducts, Preprint.

AMS subject Classification: 42B20, 42A50 Keywords: Carleson embedding theorem, vector valued functions, operator valued measures, weights.

Department of Mathematics, University of California, Berkeley, CA 94720-3840

  E-mail address: mikeg@math.berkeley.edu