A characterization of bilinear forms on the dirichlet space

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AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 7, July 2012, Pages 2429–2440 S 0002-9939(2011)11409-6

Article electronically published on November 15, 2011

ON A CHARACTERIZATION OF BILINEAR FORMS ON THE DIRICHLET SPACE

CARME CASCANTE AND JOAQUIN M. ORTEGA (Communicated by Richard Rochberg)

Abstract. _{Arcozzi, Rochberg, Sawyer and Wick obtained a} characteriza-tion of the holomorphic funccharacteriza-tions b such that the Hankel type bilinear form Tb(f, g) =

D(I + R)(f g)(z)(I + R)b(z)dv(z) is bounded onD × D, where D is

the Dirichlet space. In this paper we give an alternative proof of this charac-terization which tries to understand the similarity with the results of Mazya and Verbitsky relative to the Schr¨odinger forms on the Sobolev spaces L1

2(Rn).

1. Introduction

LetD be the Dirichlet space on the unit disk D, that is, the space of holomorphic functions onD such that

f2

D=

D|(I + R)f(z)|

2_{dv(z) < +∞,}

where Rf (z) = z_∂z∂ f (z) is the radial derivative and dv is the Lebesgue measure on

D. We recall that D coincides with the Hardy-Sobolev space H2

1 2

on the unit disk. If b is a holomorphic function onD, such that (I + R)b ∈ L1₍_{D), let T}

b denote

the bilinear form, deﬁned initially for holomorphic polynomials f, g by

Tb(f, g) =

D(I + R)(f g)(z)(I + R)b(z)dv(z). The norm of Tb is

Tb = sup{|Tb(f, g)| : f_D,g_D≤ 1}.

The object of this paper is the study of the functions b such thatTb is ﬁnite. In

fact, a characterization of the functions b satisfying this condition was given in [4], where the following theorem was shown, solving a conjecture stated by R. Rochberg:

Theorem 1.1 ([4]). If b is a holomorphic symbol onD such that (I +R)b ∈ L1_(D).

Then the following assertions are equivalent:

(i) The bilinear form Tb is bounded onD × D.

(ii) The measure dμb(z) =|(I + R)b(z)|2dv(z) is a Carleson measure for the

Dirichlet spaceD.

Received by the editors February 23, 2011.

2010 Mathematics Subject Classiﬁcation. Primary 31C25, 31C15, 47B35. Key words and phrases. Dirichlet spaces, Carleson measures.

The authors were partially supported by DGICYT Grant MTM2011-27932-C02-01, DURSI Grant 2009SGR 1303 and Grant MTM2008-02928-E.

c

2011 American Mathematical Society

Reverts to public domain 28 years from publication

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A positive measure μ on D is a Carleson measure for the Dirichlet space D if and only if there exists C > 0 such that for any f ∈ D,

D|f(z)|

2

dμ(z)≤ Cf2_D.

Stegenga (see [12]) characterized the above Carleson measures in terms of Riesz capacities. Namely, μ is a Carleson measure forD if and only if there exists C > 0 such that for any open set A on T, the unit circle, μ(T (A)) ≤ CC1

2,2(A). Here

T (A) = (_{ζ /}_∈ADα(ζ))c is the tent over A and if ζ ∈ T, Dα(θ) ={z : |1 − zζ| <

α(1− |z|2₎_{}, α > 2 is a nontangential region.}

We also recall that if A is a set on T, then the Riesz capacity is given by

C1

2,2(A) = inf{f

2

L2₍_T) : f≥ 0, I1

2[f ]≥ 1 on A}, where, if f is a nonnegative function onT, we denote for w ∈ T,

(1.1) I1 2[f ](w) = 2π 0 f (θ) |1 − we−iθ_|1 2 dθ.

The original proof of (i) implies (ii) in [4] is quite delicate and needs very accurate estimates. They use the capacitary characterization of Carleson measures, studying the relative sizes of both _V |b|2_{, where V is a certain set in} _{D, and the capacity} of the set V ∩ ∂D. Then they construct an “expanded” set, Vexp, satisfying that

V and D \ Vexp are “well separated”, but such that Vexp is not very large when

it is measured by _V exp|b

_|2 _{or by the capacity of V} _{∩ ∂D. It is then possible to} construct a pair of functions f and g such that |Tb(f, g)| can be expressed, up to

an error term, as _V |b|2_{, and to show the smallness of this error term. Some of} the estimates are obtained by working on Bergman trees and the associated tree capacities.

In the same paper, the formal similarity with a boundedness criterion for a bilin-ear form associated to the Schr¨odinger operator due to Mazja and Verbitsky, [10], is observed. Let L1

2(Rn) be the homogeneous Sobolev space obtained by complet-ing C0∞(R

n

) with respect to the quasinorm induced by the Dirichlet inner product

f, g Dir=

Rn∇f∇gdx. Given b, the Schr¨odinger form Sb on L12(Rn)× L12(Rn) is deﬁned as

Sb(f, g) =fg, b Dir.

In Corollary 2 of [10] it is shown that Sbis bounded if and only if dμb=|(−Δ)1/2b|2dx

is a Carleson measure for the space L1

2(Rn), that is, if and only if there exists C > 0 such that for any f ∈ L1

2(Rn), Rn |f|2 dμb ≤ Cf2_L1 2(Rn).

The proof of this result uses completely diﬀerent techniques to the ones used in the proof of Theorem 1.1 in [4]. An adequate integration by parts and the use of certain weights in the A2Muckenhoupt class are fundamental tools.

The main object of this paper is to give an alternative proof of Theorem 1.1, based on the existence of holomorphic potentials associated to extremal measures with respect to the capacity of a set and inspired in part in the methods of the char-acterization of the boundedness of the bilinear form Sbassociated to the Schr¨odinger

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approximate the answer to the question posed in [4] of knowing if there is an un-derlying reason for such formal similarity.

In Section 2, we recall the main properties relative to capacities and holomorphic potentials needed for the proof of the theorem. Section 3 is devoted to giving the proof of Theorem 1.1.

As usual, we will adopt the convention of using the same letter for various ab-solute constants whose values may change in each occurrence. Also, if there is no confusion, for simplicity we will write A B if there exists an absolute constant

M such that A≤ MB. We will say that two quantities A and B are equivalent if

both A B and B A, and, in that case, we will write A ≈ B.

2. Capacitary measures and holomorphic potentials

We begin this section by recalling the main properties of the capacitary extremal measure associated to an open set onT.

Theorem 2.1 (Thm. 2.2.7, [1]). If A is an open set on T, there exists a positive measure ν onT, the capacitary measure for A, such that

(i) ν is supported on A; (ii) ν = I1 2[ν] 2 2= C1 2, 2(A); (iii) P(w) := I1 2[I12[ν]](w)≤ C, ω ∈ T;

(iv) P(w) ≥ 1 on A, except for a set of capacity zero.

If we denote by |E| the Lebesgue measure of a measurable set E ⊂ T, it is proved in Theorem 20 in [11] that for any ε > 0, |E|ε C1

2,2(E). Consequently, we also have thatP(w) ≥ 1 on A, almost everywhere on T. Our next lemma gives a pointwise estimate of I1

2[I 1 2[ν]].

Lemma 2.2. There exist C, K > 0 such that for any positive measure ν onT and any eiθ_{∈ T,} (2.1) I1 2[I 1 2[ν]](e iθ )≤ C 2π 0 ln K |eiθ− eit|dν(t).

Furthermore, there exist C1, K1, δ > 0, such that for any θ, t, 0 <|eiθ− eit| ≤ δ,

(2.2) 0 < C1ln K1 |eiθ_{− e}it_| ≤ 2π 0 2π 0 dη

|eiη_{− e}it_|1₂_|eiθ_{− e}iη_|1₂.

Proof. Fubini’s theorem gives that

I1 2[I 1 2[ν]](e iθ ) = 2π 0 2π 0 dη

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Next, 2π

0

dη

|eiη− eit|12|eiθ− eiη|12 =

|eiθ_−eiη_|≤|eiθ −eit|

3

dη

|eiη_{− e}it_|1₂_|eiθ_{− e}iη_|1₂

+

|eit_−eiη_|≤|eiθ −eit|

3

dη

|eiη− eit|1₂|eiθ− eiη|1₂

+

|eiθ_−eiη_|>|eiθ −eit|

3 ,|eit−eiη|>

|eiθ −eit|

3

dη

|eiη− eit|12|eiθ− eiη|12

= I + II + III. On the region considered in the integral given by I,|eiη_{− e}it_{| ≈ |e}iθ_{− e}it_{|, and}

consequently

I 1

|eiθ− eit|1₂

|eiθ_−eiη_|≤|eiη −eit|

3

dη

|eiθ− eiη|1₂ ≈ C.

The integral given by II is estimated analogously.

Next, we split the region considered in III into two parts, one corresponding to the points where |eiη_{− e}it_{| > |e}iθ_{− e}iη_{| and one where |e}iη_{− e}it_{| ≤ |e}iθ_{− e}iη_{|. In}

the ﬁrst case the integral given by III is bounded from above by

|eiθ −eit|

3 <|eiθ−eiη|≤2

dη

|eiθ− eiη| ln

K |eiθ− eit|.

Finally, in order to obtain the estimate given in (2.2), we consider for 0 <

|eiθ_{− e}it_{| ≤ δ, where δ > 0 is to be chosen, the region where |e}iη_{− e}it_{| ≥ |e}it_{− e}iθ_|.

In that case, we have that |eiη_{− e}iθ_{| ≤ 2|e}iη_{− e}it_{|, and consequently, if δ is small}

enough,

2π 0

dη

|eiη− eit|1₂|eiθ− eiη|1₂

π |η−t|>2|t−θ| dη |η − t|  ln K1 |eiθ− eit| > 0.

If ln denotes the principal branch of the logarithm, we deﬁne the holomorphic potential of a ﬁnite positive measure μ onT by

(2.3) U(z) =

Tln

K

1− zζdμ(ζ), where K > 0 is a ﬁxed constant satisfying that lnK

2 ≥ 4π and big enough so that the estimate (2.1) in Lemma 2.2 can be applied. This choice of K > 0 implies in particular that for any z ∈ D, Re U(z) ≥ 2|Im U(z)| and |U| ≈ Re U(z). We will deﬁneV(z) = Re U(z).

We remark that if we replace K by a larger constant in the deﬁnition of the holomorphic potentialU, the estimates obtained in the following lemmas still remain true. For technical reasons which will appear later on, it will be convenient at some point to replace the constant K previously considered by 6K. When we need to specify the precise constant, we will write VK instead ofV.

Lemma 2.3. If A⊂ T is an open set, ν is the extremal measure for A and U is the holomorphic potential associated to ν, we have:

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(i) U_D C1 2,2(A)

1 2. (ii) |U(z)| ≤ C.

(iii) 1 V(z) = Re U(z) on T (A).

Proof. The proof will be based on the properties of the extremal measure ν and

the above lemma. Let us begin with (i). It is proved in Theorem 2.10 in [3] that

UD I1 2[ν]2, which, by (ii) in Theorem 2.1, coincides with C1

2,2(A) 1 2. In particular, U ∈ D = H2 1 2 ⊂ H

2_{, and consequently, it has nontangential boundary values}_U∗_almost every-where. If we denote V∗ = ReU∗, and if P (z, θ) = _|1−ze1−|z|−iθ2_|2 is the Poisson kernel, we then have |U(z)| ≈ V(z) (2.4) = 2π 0 P (z, θ)V∗(eiθ)dθ. (2.5)

Consequently, in order to prove (ii), it is enough to show thatV∗(eiθ) 1. Indeed, by (2.2) and (ii) and (iii) in Theorem 2.1, we have that V∗(eiθ) C1

22(A) + I1 2[I 1 2[ν]](e iθ₎_1.

Finally, in order to prove (iii), we observe that if z ∈ T (A), then there exists

c < 1 such that B(_|z|z , c(1− |z|)) ⊂ A. Since by (iv) in Theorem 2.1, V∗ 1 a.e.

on A, we have then that

V(z) = 2π 0 (1− |z|2₎ |z − eiθ|2V ∗_(θ)dθ ≥ B( z |z|,c(1−|z|)) (1− |z|2₎ |z − eiθ_|2V∗(θ)dθ ≥ C B( z |z|,c(1−|z|)) (1− |z|2) |z − eiθ|2dθ≈ 1 (1− |z|2₎ B(z |z|,c(1−|z|)) dθ≈ 1.

Before we state our next lemma, we give a couple of deﬁnitions. If f is a function in L2_{(D), its Bergman projection is deﬁned by}

B[f ](z) =

D

f (w)

(1− zw)2dv(w), w∈ D.

If ω is a weight onD, it was proved in [5] that B is bounded on L2_{(ω) if and only} if ω is in the class B2 deﬁned by

Deﬁnition 2.4. A weight ω is in B2if there exists C > 0 such that for any interval

I⊂ T, 1 |T (I)| T (I) ωdv 1 |T (I)| T (I) ω−1dv ≤ C. Lemma 2.5. The function V2 _{is in the B}

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Proof. Observe that

Re(U2) = (Re U)2− (Im U)2= ((ReU + Im U)(Re U − Im U) ≈ (ReU)2=V2.

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In [9] (Lemma 3 of 2.6) there is a proof of a result by Verbitsky that shows that for any β ∈ (1, +∞), the weight (I1

2[I12[ν]])

β

is in the Muckenhoupt class A1 on T, with constants independent of the measure ν. In particular, we have that the weight (V∗)2 is in the class A1onT, and consequently, it is also in the A2 class on T.

It is proved in [8] that if the weight (V∗)2 is in the Muckenhoupt class A2 onT and we consider the set Iz ={ζ ∈ T : |1 − _|z|z ζ| ≤ c(1 − |z|2)}, c < 1, then the

weight deﬁned by ˜ (V∗)2_{(z) =} 1 1− |z| Iz (V∗)2(ζ)|dζ| is in the class B2.

The estimate (2.6) gives that P [(V∗)2_]_{≈ P [Re(U}2_{)] = Re(}_U2₎_{≈ (V)}2_. Hence, in order to ﬁnish the proof of the lemma it is enough to check that

(2.7) P [(V∗)2]≈ (˜V∗)2.

If z = reiθ0 ∈ D, for any k ≥ 1, let I

k(z) ={ζ ∈ T : |1 − eiθ0ζ)| ≤ 2k(1− r)} and I0=∅. We then have P [(V∗₎2 ](z) = k≥0 Ik+1\Ik 1− |z|2 |1 − zζ|2(V ∗₎2 (ζ)dζ k≥0 1 22k₍₁_{− r)} Ik+1 (V∗)2(ζ)dζ. Since (V∗)2 _{is in A}

2, it is well known that (V∗)2 is also in A2−ε, for some ε > 0, and in particular, it satisﬁes a doubling condition of order τ < 2. Thus,

Ik+1

(V∗)2(ζ)dζ≤ C2kτ

Ik

(V∗)2(ζ)dζ. Consequently, the above sum is bounded above by k_≥0 1 2k(2−τ)₍₁− r) Iz (V∗)2(eiθ)dθ≈ (˜V∗)2(z).

The lower estimate in (2.7) is a consequence of the inequality 1 |Iz| Iz (V∗)2(ζ)dζ P [(V∗)2](z). 3. Proof of Theorem 1.1

3.1. (ii) implies (i). The proof of this implication is direct; see [4]. For the sake of completeness, we include it here. Assume that dμb(z) =|(I + R)b(z)|2dv(z) is a

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Carleson measure for the Dirichlet spaceD, and let f, g ∈ D. We then have: |Tb(f, g)| = D(I + R)f (z) g(z)(I + R)b(z)dv(z) + D(f (z) Rg(z))(I + R)b(z)dv(z) ≤ D|(I + R)f(z)| 2 dv(z) 1 2 D|g(z)| 2_{|(I + R)b(z)|}2 dv(z) 1 2 + D|f| 2_{|(I + R)b(z)|}2 dv(z) 1 2 D|Rg(z)| 2 dv(z) 1 2 fDgD.

3.2. (i) implies (ii). Assume now that Tbis bounded onD ×D. We want to prove

that dμb(z) is a Carleson measure for the Dirichlet spaceD, which is equivalent to

proving that there exists a constant K such that for any open set A⊂ T, (3.1) T (A) |(I + R)b|2 (z)dv(z)≤ KC1 2,2(A). We begin with some technical lemmas needed in the proof.

Lemma 3.1. If Tbis bounded onD ×D, the symbol b belongs to the Dirichlet space

D.

Proof. Let h∈ D. By hypothesis, |

D(I + R)h(z)(I + R)b(z)dv(z)| TbhD.

Duality gives then that b∈ D and bD Tb.

Lemma 3.2. Let b ∈ D and f be the holomorphic function on D deﬁned by f =

(I + R)−1B[[(I + R)b]χT (A)], where B is the Bergman projection onD. Then f ∈ D

andfD bD.

Proof. We have that (I + R)f = B[[(I + R)b]χT (A)], and consequently, the

bound-edness of the Bergman projection on L2(D) and Lemma 3.1 give that

fD=B[[(I + R)b]χT (A)]L2₍_D) (I + R)bχ_{T (A)}_L2₍_D)≤ b_D. We now proceed to prove the theorem. Given an open set A, let ν be the extremal measure as in Theorem 2.1, let U ∈ D be the holomorphic potential associated to

ν and A, deﬁned in (2.3), and let f be the holomorphic function associated to b,

deﬁned in Lemma 3.2. Next, observe that

Tb(U, f U) = D(I + R)f (z)(I + R)b(z)dv(z) =

DB[[(I + R)b]χT (A)](z)(I + R)b(z)dv(z) =

D[(I + R)b]χT (A)(z)(I + R)b(z)dv(z) = T (A) |(I + R)b(z)|2 dv(z) = μb(T (A)). (3.2)

Since Tb is bounded, by Lemma 2.3 we have

|Tb(U, f U)| UD f UD C1 22(A) 1 2f UD.

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We will show that (3.3) f UD T (A) |(I + R)b(z)|2 dv(z) 1 2 = μb(T (A)) 1 2. Indeed, if (3.3) holds, then by (3.2), μb(T (A)) C1

2, 2(A) 1

2μ_b(T (A)) 1

2, and since by Lemma 3.1, μb(T (A)) bD < +∞, we ﬁnish the proof of (i) implies (ii).

So, we are led to show (3.3). But (I + R)_Uf = (I+R)f_U −f R_U2U. Thus it is enough to show that (3.4) D |(I + R)f(z)|2 |U2_(z)_| dv(z) μb(T (A)) and (3.5) D |f(z)RU(z)|2 |U4_(z)| dv(z) μb(T (A)).

We begin with (3.4). Since |U| ≈ V and Lemma 2.5 gives that _V12 is a weight in

where in the last estimate we have used that by Lemma 2.3,V2_(z)_{≥ C on T (A).} In order to prove the estimate (3.5), we need some technical results, which we state in the following two lemmas.

Lemma 3.3. There exists C > 0 such that for any compactly supported real-valued C1 _{function φ and any nonidentically zero positive measure ν supported on}_{T, if U}

is the holomorphic potential deﬁned a.e. in C as in Lemma 2.3 and V = Re U, we have that (3.6) Cφ(z) 2|∇V(z)|2 (V(z))4 dv(z)≤ C C |∇φ(z)|2 (V(z))2 dv(z).

Proof. In order to prove the lemma, we will make the following reduction.

Let us consider the measures νε= μ∗ϕε, regularizations of ν, where ϕ is a radial

C∞_{function on}_{C, which is zero outside a neighborhood of 0 and such that} Cϕ = 1 and ϕε(y) = 1_εϕ(y/ε), for any y ∈ C. It is enough to show the estimate (3.6) for

the functions Vε (z) = Cln K |1 − zζ|dνε(ζ),

with constants independent of ε > 0. Indeed, since νε converges weakly to ν, as

ε → 0, we have that both Vε _and _∇Vε _{converge pointwise almost everywhere to}

V and ∇V respectively as ε → 0. Then applying Fatou’s lemma ﬁrst and the

Dominated Convergence Theorem (since by deﬁnition, _V1ε ≤ C for any ε > 0), we have Cφ(z) 2|∇V(z)|2 (V(z))4 dv(z)≤ lim inf_ε_→0 Cφ(z) 2|∇Vε(z)|2 (Vε_(z))4 dv(z) lim inf ε→0 C |∇φ(z)|2 (Vε_(z))2dv(z) = C |∇φ(z)|2 (V(z))2 dv(z).

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So we are left to show (3.7) C φ(z)2|∇V ε_(z)_|2 (Vε_(z))4 dv(z) C |∇φ(z)|2 (Vε_(z))2dv(z),

with constants independent of ε > 0. Applying integration by parts and using that the function φ is compactly supported, we then have

Cφ(z) 2|∇Vε(z)|2 (Vε_(z))4 dv(z) = − Cφ(z) 2ΔV ε_(z) (Vε_(z))3dv(z)− 2 Cφ(z)∇φ(z) · ∇V ε (z) 1 Vε_(z)3dv(z) + 4 Cφ(z) 2|∇V ε (z)|2 (Vε_(z))4 dv(z).

Hence, since−ΔVε(z) = νεis a positive measure,

3 C φ(z)2|∇V ε_(z)_|2 (Vε_(z))4 dv(z) = 2 Cφ(z)∇φ(z) · ∇V ε (z) 1 Vε_(z)3dv(z) + Cφ(z) 2 ΔV ε (z) (Vε_(z))3dv(z) ≤ 2 Cφ(z)∇φ(z) · ∇V ε (z) 1 Vε_(z)3dv(z).

Applying H¨older’s inequality to the last estimate, we obtain 3 Cφ(z) 2|∇Vε(z)|2 (Vε_(z))4 dv(z) ≤ 2 C |∇φ(z)|2 (Vε_(z))2dv(z) 1 2 Cφ(z) 2|∇Vε(z)|2 (Vε_(z))4 dv(z) 1 2 ,

which gives that Cφ(z) 2|∇Vε(z)|2 (Vε_(z))4 dv(z)≤ 2 3 2 C |∇φ(z)|2 (Vε_(z))2dv(z),

and that gives (3.7) and ends the lemma.

The next lemma is a modiﬁcation of a classical extension theorem on real Sobolev spaces (see [6]) to a weighted situation. We construct an extension to the whole plane of a function in the Sobolev space on the disc with respect to the weight

dv(z)

V(z)2, with estimates of the corresponding norm. We recall that if L > 0, we will writeVL(z) =

Tln

L

|1 − zζ|dν(ζ) when we want to specify the constant L. Namely,

we have:

Lemma 3.4. There exists C > 0, such that for any real-valuedC1_{function φ on}_D,

there exists a real-valued compactly supported C1 _{extension to}_{C of φ, ˜φ, satisfying}

that (3.8) C |∇ ˜φ(z)|2 V6K(z)2 dv(z)≤ C D |∇φ(z)|2 V2K(z)2 dv(z) + D |φ(z)|2 V2K(z)2 dv(z) .

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Proof. We ﬁrst extend the function φ to a neighborhood of the closed unit disc by

deﬁning the function

φe(ρeiθ) = 3φ((2− ρ)eiθ)− 2φ((3 − 2ρ)eiθ), 1 < ρ < 3₂. We observe that lim

ρ→1φ e

(ρeiθ) = φ(eiθ), and lim

ρ→1 ∂φe ∂ρ (ρe iθ ) =∂φ ∂ρ(e iθ ), and consequently, φe _{is a} _C1 _{extension of φ to} _{|z| <} 3

2. Let 0 < ε < 1

4 be ﬁxed and let ψ be a C∞ function onC such that 0 ≤ ψ ≤ 1, ψ(z) = 1 for |z| < 1 + ε, and ψ(z) = 0 for |z| > 3₂ − ε. We consider the function ˜φ = ψφe_{, which is a} _C1 compactly supported function.

We have C |∇ ˜φ(z)|2 V6K(z)2 dv(z) |z|<3 2 |∇φe (z)|2 V6K(z)2 dv(z) + |z|<3 2 |φe (z)|2 V6K(z)2 dv(z) = I + II.

In order to estimate the integrals in both I and II, we split both integrals into two pieces, one corresponding to D, and the other corresponding to 1 ≤ |z| < 3₂. In this last region, we will apply an appropriate change of variables to transform the integrals into integrals over a subset ofD. We ﬁrst observe that for any 0 < r < 1, 1−r ≤ |reiθ_{−ζ| and |e}iθ_{−ζ| ≤ 2|re}iθ_{−ζ|. Consequently, for 1 < ρ <} 3

2,|ρe

iθ_{−ζ| ≤}

|1 − (2 − ρ)| + |eiθ_{− ζ| ≤ 3|(2 − ρ)e}iθ_{− ζ|. Similarly, |ρe}iθ_{− ζ| ≤} 5

2|(3 − 2ρ)e iθ_{− ζ|.}

Hence, we have that both ln_|ρe6Kiθ_−ζ| ≥ ln 2K

|(2−ρ)eiθ_−ζ| ≥ 1 and ln 6K |ρeiθ_−ζ| ≥

ln 2K

|(3−2ρ)eiθ_−ζ| ≥ 0, and consequently, 1<_|z|<3 2 |∇φe_(z)_|2 V3K(z)2 dv(z) 1<|z|<3 2 |∇φ((2 − |z|)z |z|)|2 V6K(z)2 dv(z) + 1<|z|<3 2 |∇φ((3 − 2|z|)z |z|)|2 V6K(z)2 dv(z) 1<|z|<3 2 |∇φ((2 − |z|)z |z|))|2 V2K((2− |z|)_|z|z))2 dv(z) + 1<|z|<3 2 |∇φ((3 − 2|z|)z |z|))|2 V2K((3− 2|z|)_|z|z))2 dv(z) D |∇φ(z)|2 V2K(z)2 dv(z).

This gives that

I D |∇φ(z)|2 V2K(z)2 dv(z).

Next we deal with the integral given by II. Arguing as before, we have that 1<_|z|<3 2 |φe_(z)_|2 V6K(z)2 dv(z) D |φ(z)|2 V2K(z)2 dv(z).

Theorem 3.5. There exists a constant C > 0 such that for any h in the Dirichlet spaceD, D|h(z)| 2|RV(z)| 2 V6K(z)4 dv(z)≤ C D |(I + R)h(z)|2 VK(z)2 dv(z).

Proof. SinceU is holomorphic, |RU| ≤ |∇V|. If ρ < 1, and hρ(z) = h(ρz), we apply

Lemmas 3.3 and 3.4 to Re hρ and Im hρ, and we obtain that

D |hρ(z)RV(z)|2 V6K(z)4 dv(z) D |∇hρ(z)|2 V2K(z)2 dv(z).

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Next we have that VK(z) V2K(z) and consequently, by Fatou’s lemma, D|h(z)| 2|∇V(z)|2 V6K(z)4 dv(z)≤ lim inf ρ→1 D|hρ(z)| 2|∇V(z)|2 V6K(z)4 dv(z) lim inf ρ→1 D |∇hρ(z)|2 V2K(z)2 dv(z) D |∇h(z)|2 VK(z)2 dv(z).

Finally, the pointwise estimate in Lemma 3.6 in [2] gives that if α < β, there exists C > 0 such that for any ζ ∈ T,

Dα(ζ) |∇h(z)|2 (1− |z|2)−1dv(z) Dβ(ζ) |Rh(z)|2 (1− |z|2)−1dv(z).

Consequently, multiplying by 1/(VK∗)2(ζ), integrating and applying Lemma 2.7 to

the weight 1/(VK∗)2(ζ) (which is also in the class A2), we easily obtain D |∇h(z)|2 VK(z)2 dv(z) D |Rh(z)|2 VK(z)2 dv(z).

Finally, a standard argument using integral representations (see, for instance, Theorem 2.8 in [7]) gives that the right-hand side of the above integral is bounded up to a constant by D |Rh(z)|2 VK(z)2 dv(z) D |(I + R)h(z)|2 VK(z)2 dv(z).

Now we can ﬁnish the proof of (i) implies (ii), proving the estimate (3.5). Since

|U| V, Theorem 3.5 and the argument used in the proof of (3.4) give the proof.

References

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in the unit ball ofCn_{. Revista Matem´}_{atica Iberoamericana 4 (1988), 123–152. MR1009122} (90h:32011)

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[4] N. Arcozzi, R. Rochberg, E. Sawyer and B.D. Wick: Bilinear forms on the Dirichlet space, Analysis & PDE 3 (2010), no. 1, 21-47. MR2663410 (2011f:31011)

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[7] C. Cascante and J.M. Ortega: Carleson measures for weighted Hardy-Sobolev spaces. Nagoya Math. J. 186 (2007), 29–68. MR2334364 (2008g:32011)

[8] C. Cascante and J.M. Ortega: Carleson measures for weighted holomorphic Besov spaces, Ark. Mat. 49 (2011), 31–59. MR2784256

[9] V.G. Mazya and T.O. Shaposhnikova: Theory of multipliers in spaces of diﬀerentiable func-tions, Monographs and Studies in Mathematics 23, Pitman, 1985. MR785568 (87j:46074) [10] V.G. Mazya and I.E. Verbitsky: The Schr¨odinger operator on the energy space: boundedness

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[11] N.G. Meyers: A theory of capacities for potentials of functions in Lebesgue classes. Math. Scand. 26 (1971), 255–292. MR0277741 (43:3474)

[12] D. Stegenga: Multipliers of the Dirichlet space. Illinois J. Math. 24 (1980), 113–139. MR550655 (81a:30027)

Departmento Matem`atica Aplicada i An`alisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain

E-mail address: cascante@ub.edu

Departmento Matem`atica Aplicada i An`alisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain