CARLESON POTENTIALS AND THE REPRODUCING KERNEL THESIS

FOR EMBEDDING THEOREMS

STEFANIE PETERMICHL

1

, SERGEI TREIL

2

, AND BRETT D. WICK

3

Abstract. In this note we present a new proof of the Carleson Embedding Theorem on the unit

disc and unit ball in C

n

The only technical tool used in the proof of this fact is Green’s formula. The .

starting point is that every Carleson measure gives rise to a bounded sub-harmonic function. Using

this function we construct a new related Carleson measure that allows for a simple embedding.

In the case of the disc D this gives the best known constant, with the previous best given by

N. Nikolskii.

0. Introduction

The famous Carleson Embedding Theorem for the unit disc states, in particular, that the embedding of the Hardy space H2

into a space L

2

(µ) can be checked on reproducing kernels of the

Hardy space. Namely, it can be stated as follows:

Theorem 0.1 (Carleson Embedding Theorem). Let µ be a non-negative measure in D. Then the

following are equivalent:

(i) The Hardy space H2

(D) is embedded in L

2

(µ), i.e.

Z

D

|f(z)|

2

dµ(z) ≤ A

2

kfk

2

H2

(D)

∀f ∈ H2

(D).

(ii)

C(µ)

2

:= sup

z∈D

kkzk

2

L2

(µ) = sup

z∈D

kPzkL1

(µ) < ∞,

where kz(ξ) =

(1−|z|

2

(

1/2

1−ξz

, is the normalized reproducing kernel for the Hardy space H2

(D).

(iii)

I(µ) := sup

1

r

µ(D ∩ Q(ξ, r)) : r > 0, ξ ∈ T

,∞ >

where Q(ξ, r) is a ball in C with center ξ on T and radius r.

Moreover, the best possible constant A2

in (i), the constants C(µ)

2

and I(µ) are equivalent in

the sense of two-sided estimates.

Property (iii) is typically taken as the definition of a Carleson measure on D. Condition (ii)

can be considered as a conformally invariant definition of a Carleson measure. The equivalence (ii)

⇐⇒ (iii) above is a simple and standard fact that can be obtained by integrating |kz(·)|

2

using

its distribution function. The condition (ii) means that we are checking embedding only on the

reproducing kernels, and not on all H2

(D) functions. Thus the implication (i) =⇒ (ii), as well as

the estimate C ≤ A are trivial, so the only non-trivial estimate in this theorem is (ii) =⇒ (i).

Results of such type, saying that it is sufficient to check the boundedness of an operator on

reproducing kernels are called “reproducing kernel thesis”.

1991 Mathematics Subject Classification. 30D55,30E05,32A53,42B30,46E22.

1. Research supported in part by a National Science Foundation Grant.

2. Research supported in part by a National Science Foundation Grant.

3. Research supported in part by a National Science Foundation Grant RTG Grant to Vanderbilt University.

12 STEFANIE PETERMICHL, SERGEI TREIL, AND BRETT D. WICK

In this note we present a new simple proof of the implication (ii) =⇒ (i) which is quite different

from the known ones. This proof also gives the best known estimate of the norm of the embedding

operator. Namely, we present a simple proof of the following theorem:

Theorem 0.2. Suppose

sup

λ∈supp µ

Z

D

|kλ(z)|

2

dµ(z) =: A < ∞.

Then Z

D

|f(z)|

2

dµ(z) ≤ 2eAkfk

2

H2

(D)

, ∀f ∈ H2

.

Recall that the Hardy space can be defined as the closure of analytic polynomials in L

2

(T,m),

where m is normalized (m(T) = 1) Lebesgue measure on T = ∂D with the norm inherited from

L

2

(T,m). The elements of H2

admit natural analytic continuation inside the unit disc D (see [3]),

so the integral in the theorem

R

D

|f(z)|

2

dµ(z) is defined.

Note that the theorem says that it is sufficient to check the embedding not on all reproducing

kernels kλ, but only on kλ, λ ∈ supp µ. This fact was known before, cf. [6, p. 151], but the constant

2e is the best known to date. In [6, p. 151] the constant 32 in the reproducing kernel thesis was

obtained, and later in [5, p. 105] the constant was improved to 16.

The proof we are going to present is a simple, “conformally invariant” proof, with the main tool

used being Green’s formula. This proof generalizes easily to the unit ball in C

n

.

Recall, that the Hardy space H2

(Bn) on the unit ball in C

n

can be defined as the closure of

polynomials in L

2

(S, σ), where σ is the Lebesgue measure on the boundary Sn = ∂Bn (see [4], or

[7] for other equivalent definitions, as well as for more information about this space).

We get the following “reproducing kernel thesis” for H2

(Bn).

Theorem 0.3. Suppose

sup

λ∈supp µ

Z

Bn

|kλ(z)|

2

dµ(z) = C,

where kλ, λ ∈ Bn is the normalized reproducing kernel of H2

(Bn). Then

Z

Bn

|f(z)|

2

dµ(z) ≤ e

(2n)!

(n!)

2

Ckfk

2

H2

(Bn)

, ∀f ∈ H2

(Bn).

Remark 0.4. The statement of the theorem does not depend on the choice of normalization of the

measure σ because if one replaces σ by cσ one would need to multiply the reproducing kernel by

c

−1/2

Usually normalization is chose by assuming that σ(Bn) = 1, and in this case the reproducing .

kernel kλ is given by (see [7])

kλ(z) =

(1 − |λ|

2

(

n/2

(1 − hz, λi)

n

,

with h·, ·i denoting the standard Hermitian inner product in C

n

.

Remark 0.5. The theorem is well known, and is usually proved using real variable methods. A new

part here would be the estimate and the fact that it is sufficient to check embedding not on all

reproducing kernels, but only on kλ, λ ∈ supp µ. We do not see how to immediately get the latter

from known results, short of repeating the proof of the Carleson Embedding Theorem given in [5]

in the context of the unit ball.

The authors would like to thank Alexander Volberg and Dechao Zheng for useful discussions.

Throughout the paper the notation := means equal by definition, and A . B means there exists

an absolute positive constant C such that A ≤ CB. The expression A ≈ B means A . B and

B . A both hold.CARELSON POTENTIALS AND EMBEDDING THEOREMS 3

1. The Embedding Theorem for the Unit Disc D

1.1. Uchiyama’s Lemma. We need the following Lemma, a version of which was probably first

proved by Uchiyama.

Lemma 1.1. Suppose that ϕ ≤ 0 is a subharmonic function. Then

dν(z) :=

1

2π

e

ϕ∆ϕ(z) log

1

|z|

dA(z)

is a Carleson measure and the embedding H2 ⊂ L

2

(ν) is a contraction. More precisely, for any

f ∈ H2

(D) we have

Z

D

|f(z)|

2

dν(z) ≤ kfk

2

H2

(D)

.

Proof. To prove this lemma we will simply use Green’s Formula applied to a particular function.

First, recall that Green’s Formula for a function u says

1

2π

Z

D

∆u(z) log

1

|z|

dA(z) =

Z

T

u(ξ)dm(ξ) − u(0),

where m is the normalized (m(T) = 1) Lebesgue measure on the unit circle T = ∂D.

We now let u = e

ϕ

|f|

2

Let us compute the Laplacian of this function. Recalling the definition .

of ∂- and ¯∂-derivatives,

∂f =

1

2

∂f

∂x

− i

∂f

∂y

λj

1 − hz, λi

−

zj

1 − |z|

2

Pz(λ).

Using that Pz(λ) is real valued and ∂jH = ¯∂jH for any function H we have

¯∂jPz(λ) = n

λj

1 − hλ, zi

−

zj

1 − |z|

2

Pz(λ).

Combining this we find that

¯∂i∂jPz(λ)

= nPz(λ)

n

λj

1 − hz, λi

−

zj

1 − |z|

2

brett.d.wick@vanderbilt.edu

Deontic logic concerns normative propositions, such as those used in law, engineering specifications, and computer programs. In other words, propositions that are translations of commands or "ought" or "must (not)" statements in ordinary language. The deontic character of such logic requires formalism that extends the first-order predicate calculus. Representative of this is the tool KED.[4]

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