On a Characterization of Bilinear Forms on the Dirichlet Space

Home , Bilinear form

PROCEEDINGS OF THE 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 characterization of the holomorphic functions b such that the Hankel type bilinear form Tb(f,g)= D(I + R)(fg)(z)(I + R)b(z)dv(z) is bounded on D×D,whereD is the Dirichlet space. In this paper we give an alternative proof of this characterization which tries to understand the similarity with the results of Mazya 1 Rn and Verbitsky relative to the Schr¨odinger forms on the Sobolev spaces L2( ).

1. Introduction Let D be the Dirichlet space on the unit disk D, that is, the space of holomorphic D functions on such that 2 2 fD = |(I + R)f(z)| dv(z) < +∞, D ∂ where Rf(z)=z ∂z f(z) is the radial derivative and dv is the Lebesgue measure on 2 D. We recall that D coincides with the Hardy-Sobolev space H 1 on the unit disk. 2 1 If b is a holomorphic function on D, such that (I + R)b ∈ L (D), let Tb denote the bilinear form, deﬁned initially for holomorphic polynomials f,g by

Tb(f,g)= (I + R)(fg)(z)(I + R)b(z)dv(z). D

The norm of Tb is

Tb =sup{|Tb(f,g)| : fD, gD ≤ 1}.

The object of this paper is the study of the functions b such that Tb 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 on D such that (I +R)b ∈ L1(D). Then the following assertions are equivalent:

(i) The bilinear form Tb is bounded on D×D. 2 (ii) The measure dμb(z)=|(I + R)b(z)| dv(z) is a Carleson measure for the Dirichlet space D.

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 2429

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2430 CARME CASCANTE AND JOAQUIN M. ORTEGA

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, 2 2 |f(z)| dμ(z) ≤ CfD. D Stegenga (see [12]) characterized the above Carleson measures in terms of Riesz capacities. Namely, μ is a Carleson measure for D if and only if there exists C>0 T ≤ such that for any open set A on , the unit circle, μ(T (A)) CC1 ,2(A). Here 2 c ∈ T { | − | T (A)=( ζ/∈A Dα(ζ)) is the tent over A and if ζ , 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 2 C 1 (A) = inf{f 2 T : f ≥ 0,I1 [f] ≥ 1onA}, 2 ,2 L ( ) 2 where, if f is a nonnegative function on T, we denote for w ∈ T, 2π f(θ) (1.1) I 1 [f](w)= 1 dθ. 2 −iθ 0 |1 − we | 2 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 | |2 D the relative sizes of both V b ,whereV is a certain set in , and the capacity of the set V ∩ ∂D. Then they construct an “expanded” set, Vexp, satisfying that D \ V and Vexp are “well separated”, but such that Vexp is not very large when it is measured by |b|2 or by the capacity of V ∩ ∂D. It is then possible to Vexp construct a pair of functions f and g such that |Tb(f,g)| can be expressed, up to | |2 an error term, as V b , 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 bilinear form associated to the Schrödinger operator due to Mazja and Verbitsky, [10], 1 Rn is observed. Let L2( ) be the homogeneous Sobolev space obtained by complet- ∞ Rn ing C0 ( ) with respect to the quasinorm induced by the Dirichlet inner product ∇ ∇ 1 Rn × 1 Rn f,g Dir = Rn f gdx. Given b, the Schrödinger form Sb on L2( ) L2( )is defined as

Sb(f,g)=fg,b Dir. 1/2 2 In Corollary 2 of [10] it is shown that Sb is bounded if and only if dμb =|(−Δ) b| dx 1 Rn is a Carleson measure for the space L2( ), that is, if and only if there exists C>0 such that for any f ∈ L1(Rn), 2 | |2 ≤ 2 f dμb C f L1(Rn). Rn 2 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 A2 Muckenhoupt 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 characterization of the boundedness of the bilinear form Sb associated to the Schr¨odinger operator obtained in [10]. This approach is technically much simpler and tries to

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BILINEAR FORMS ON THE DIRICHLET SPACE 2431

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 absolute 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 on T.

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

(i) ν is supported on A; 2 (ii) ν = I 1 [ν] = C 1 (A); 2 2 2 , 2 (iii) P(w):=I 1 [I 1 [ν]](w) ≤ C, ω ∈ T; 2 2 (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,itis ε proved in Theorem 20 in [11] that for any ε>0, |E| C 1 (E). Consequently, 2 ,2 we also have that P(w) ≥ 1onA, almost everywhere on T. Our next lemma gives a pointwise estimate of I 1 [I 1 [ν]]. 2 2

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

iθ it Furthermore, there exist C1,K1,δ >0, such that for any θ, t, 0 < |e − e |≤δ, 2π 2π K1 dη (2.2) 0

Proof. Fubini’s theorem gives that 2π 2π iθ dη I 1 [I 1 [ν]](e )= 1 1 dν(t). 2 2 iη it iθ iη 0 0 |e − e | 2 |e − e | 2

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2432 CARME CASCANTE AND JOAQUIN M. ORTEGA

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BILINEAR FORMS ON THE DIRICHLET SPACE 2433

1 (i) UD C 1 (A) 2 . 2 ,2 (ii) |U(z)|≤C. (iii) 1 V(z)=ReU(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 I 1 [ν]2, 2

1 which, by (ii) in Theorem 2.1, coincides with C 1 (A) 2 .Inparticular,U∈D= 2 ,2 2 2 ∗ H 1 ⊂ H , and consequently, it has nontangential boundary values U almost every- 2 V∗ U ∗ 1−|z|2 where. If we denote =Re ,andifP (z,θ)= |1−ze−iθ |2 is the Poisson kernel, we then have (2.4) |U(z)|≈V(z) 2π (2.5) = P (z,θ)V∗(eiθ)dθ. 0 Consequently, in order to prove (ii), it is enough to show that V∗(eiθ) 1. Indeed, ∗ iθ by (2.2) and (ii) and (iii) in Theorem 2.1, we have that V (e ) C 1 (A)+ 2 2 iθ I 1 [I 1 [ν]](e ) 1. 2 2 Finally, in order to prove (iii), we observe that if z ∈ T (A), then there exists z −| | ⊂ V∗ c<1 such that B( |z| ,c(1 z )) A. Since by (iv) in Theorem 2.1, 1 a.e. on A, we have then that 2π (1 −|z|2) V(z)= V∗(θ)dθ |z − eiθ|2 0 (1 −|z|2) ≥ V∗(θ)dθ | − iθ|2 B( z ,c(1−|z|)) z e |z| −| |2 ≥ (1 z ) ≈ 1 ≈ C iθ 2 dθ 2 dθ 1. z −| | |z − e | (1 −|z| ) z −| | B( |z| ,c(1 z )) B( |z| ,c(1 z ))

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 f(w) ∈ D B[f](z)= 2 dv(w),w . D (1 − zw) If ω is a weight on D, it was proved in [5] that B is bounded on L2(ω)ifandonly if ω is in the class B2 defined by

Deﬁnition 2.4. Aweightω is in B2 if there exists C>0 such that for any interval I ⊂ T, 1 1 −1 ≤ | | ωdv | | ω dv C. T (I) T (I) T (I) T (I)

2 Lemma 2.5. The function V is in the B2 class.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2434 CARME CASCANTE AND JOAQUIN M. ORTEGA

Proof. Observe that

Re(U 2)=(ReU)2 − (Im U)2 = ((Re U +ImU)(Re U−Im U) ≈ (2.6) (Re U)2 = V2.

In [9] (Lemma 3 of 2.6) there is a proof of a result by Verbitsky that shows that β for any β ∈ (1, +∞), the weight (I 1 [I 1 [ν]]) is in the Muckenhoupt class A1 on 2 2 T, with constants independent of the measure ν. In particular, we have that the ∗ 2 weight (V ) is in the class A1 on T, and consequently, it is also in the A2 class on T. ∗ 2 It is proved in [8] that if the weight (V ) is in the Muckenhoupt class A2 on T { ∈ T | − z |≤ −| |2 } and we consider the set Iz = ζ : 1 |z| ζ c(1 z ) , c<1, then the weight deﬁned by 1 V˜∗ 2 V∗ 2 | | ( ) (z)= −| | ( ) (ζ) dζ 1 z Iz

is in the class B2. The estimate (2.6) gives that P [(V∗)2] ≈ P [Re(U 2)] = Re(U 2) ≈ (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.

iθ iθ k If z = re 0 ∈ D, for any k ≥ 1, let Ik(z)={ζ ∈ T : |1 − e 0 ζ)|≤2 (1 − r)} and I0 = ∅.Wethenhave 1 −|z|2 P [(V∗)2](z)= (V∗)2(ζ)dζ 2 I \I |1 − zζ| k≥0 k+1 k 1 (V∗)2(ζ)dζ. 22k(1 − r) k≥0 Ik+1

∗ 2 ∗ 2 2−ε Since (V ) is in A2,itiswellknownthat(V ) is also in A ,forsomeε> 0, and in particular, it satisﬁes a doubling condition of order τ<2. Thus, (V∗)2(ζ)dζ ≤ C2kτ (V∗)2(ζ)dζ. Consequently, the above sum is bounded Ik+1 Ik above by 1 (V∗)2(eiθ)dθ ≈ (V˜∗)2(z). 2k(2−τ)(1 − r) k≥0 Iz The lower estimate in (2.7) is a consequence of the inequality 1 V∗ 2 V∗ 2 | | ( ) (ζ)dζ P [( ) ](z). Iz Iz

3. Proof of Theorem 1.1 3.1. (ii) implies (i). The proof of this implication is direct; see [4]. For the sake 2 of completeness, we include it here. Assume that dμb(z)=|(I + R)b(z)| dv(z)isa

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BILINEAR FORMS ON THE DIRICHLET SPACE 2435

Carleson measure for the Dirichlet space D, and let f,g ∈D.Wethenhave: |T (f,g)| b = (I + R)f(z) g(z)(I + R)b(z)dv(z)+ (f(z) Rg(z))(I + R)b(z)dv(z) D D 1 1 2 2 ≤ |(I + R)f(z)|2dv(z) |g(z)|2|(I + R)b(z)|2dv(z) D D 1 1 2 2 2 2 2 + |f| |(I + R)b(z)| dv(z) |Rg(z)| dv(z) fDgD. D D

3.2. (i) implies (ii). Assume now that Tb is bounded on D×D.Wewanttoprove that dμb(z) is a Carleson measure for the Dirichlet space D, which is equivalent to proving that there exists a constant K such that for any open set A ⊂ T, 2 (3.1) |(I + R)b| (z)dv(z) ≤ KC1 (A). 2 ,2 T (A) We begin with some technical lemmas needed in the proof.

Lemma 3.1. If Tb is bounded on D×D, the symbol b belongs to the Dirichlet space D. Proof. Let h ∈D. By hypothesis,

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

Duality gives then that b ∈Dand bD Tb. Lemma 3.2. Let b ∈Dand f be the holomorphic function on D deﬁned by f = −1 (I +R) B[[(I +R)b]χT (A)],whereB is the Bergman projection on D.Thenf ∈D and fD bD.

Proof. We have that (I + R)f = B[[(I + R)b]χT (A)], and consequently, the boundedness of the Bergman projection on L2(D) and Lemma 3.1 give that fD = B[[(I + R)b]χT (A)]L2(D) (I + R)bχT (A)L2(D) ≤bD. We now proceed to prove the theorem. Given an open set A,letν be the extremal measure as in Theorem 2.1, let U∈Dbe 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 U f Tb( , U )= (I + R)f(z)(I + R)b(z)dv(z) D

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

Since Tb is bounded, by Lemma 2.3 we have

f f 1 f |Tb(U, )| UD D C 1 (A) 2 D. U U 2 2 U

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2436 CARME CASCANTE AND JOAQUIN M. ORTEGA

We will show that 1 2 f 1 | |2 2 (3.3) U D (I + R)b(z) dv(z) = μb(T (A)) . T (A)

1 1 Indeed, if (3.3) holds, then by (3.2), μb(T (A)) C 1 (A) 2 μb(T (A)) 2 , and since 2 , 2 by Lemma 3.1, μb(T (A)) bD < +∞, we finish the proof of (i) implies (ii). f (I+R)f − fRU So, we are led to show (3.3). But (I + R) U = U U 2 .Thusitisenough to show that | |2 (I + R)f(z) (3.4) 2 dv(z) μb(T (A)) D |U (z)| and | U |2 f(z)R (z) (3.5) 4 dv(z) μb(T (A)). D |U (z)| |U| ≈ V 1 We begin with (3.4). Since and Lemma 2.5 gives that V2 is a weight in B2, the integral corresponding to the left-hand side of (3.4) is bounded by |(I + R)f(z)|2 |[B[(I + R)b]χ ](z)|2 dv(z) ≤ T (A) dv(z) |U 2(z)| V2(z) D D |(I + R)b(z)|2 dv(z) |(I + R)b(z)|2dv(z)=μ (T (A)), V2 b T (A) (z) T (A) 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,ifU is the holomorphic potential defined a.e. in C as in Lemma 2.3 and V =ReU,we have that |∇V |2 |∇ |2 2 (z) ≤ φ(z) (3.6) φ(z) 4 dv(z) C 2 dv(z). C (V(z)) C (V(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,whereϕ is a radial C∞ C function on , which is zero outside a neighborhood of 0 and such that C ϕ =1 1 ∈ C and ϕε(y)= ε ϕ(y/ε), for any y . It is enough to show the estimate (3.6) for the functions ε K V (z)= ln dνε(ζ), C |1 − zζ| 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 1 ≤ Dominated Convergence Theorem (since by definition, Vε C for any ε>0), we have |∇V(z)|2 |∇Vε(z)|2 φ(z)2 dv(z) ≤ lim inf φ(z)2 dv(z) (V(z))4 ε→0 (Vε(z))4 C C |∇ |2 |∇ |2 φ(z) φ(z) lim inf ε 2 dv(z)= 2 dv(z). ε→0 C (V (z)) C (V(z))

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BILINEAR FORMS ON THE DIRICHLET SPACE 2437

So we are left to show |∇Vε |2 |∇ |2 2 (z) φ(z) (3.7) φ(z) ε 4 dv(z) ε 2 dv(z), C (V (z)) C (V (z)) with constants independent of ε>0. Applying integration by parts and using that the function φ is compactly supported, we then have |∇Vε(z)|2 φ(z)2 dv(z) (Vε(z))4 C ΔVε(z) 1 = − φ(z)2 dv(z) − 2 φ(z)∇φ(z) ·∇Vε(z) dv(z) (Vε(z))3 Vε(z)3 C C |∇Vε |2 2 (z) +4 φ(z) ε 4 dv(z). C (V (z)) ε Hence, since −ΔV (z)=νε is a positive measure, |∇Vε(z)|2 3 φ(z)2 dv(z) (Vε(z))4 C 1 ΔVε(z) =2 φ(z)∇φ(z) ·∇Vε(z) dv(z)+ φ(z)2 dv(z) Vε 3 Vε 3 C (z) C ( (z)) ≤ ∇ ·∇Vε 1 2 φ(z) φ(z) (z) ε 3 dv(z). C V (z) Applying H¨older’s inequality to the last estimate, we obtain |∇Vε |2 2 (z) 3 φ(z) ε 4 dv(z) C (V (z)) 1 1 |∇ |2 2 |∇Vε |2 2 ≤ φ(z) 2 (z) 2 ε 2 dv(z) φ(z) ε 4 dv(z) , C (V (z)) C (V (z)) which gives that |∇Vε |2 2 |∇ |2 2 (z) ≤ 2 φ(z) φ(z) ε 4 dv(z) ε 2 dv(z), C (V (z)) 3 C (V (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 L write VL(z)= ln dν(ζ) when we want to specify the constant L.Namely, T |1 − zζ| we have: Lemma 3.4. There exists C>0, such that for any real-valued C1 function φ on D, there exists a real-valued compactly supported C1 extension to C of φ, φ˜, satisfying that |∇ ˜ |2 |∇ |2 | |2 φ(z) ≤ φ(z) φ(z) (3.8) 2 dv(z) C 2 dv(z)+ 2 dv(z) . C V6K (z) D V2K (z) D V2K (z)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2438 CARME CASCANTE AND JOAQUIN M. ORTEGA

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), ∂φe ∂φ 1 <ρ< 3 . We observe that lim φe(ρeiθ)=φ(eiθ), and lim (ρeiθ)= (eiθ), 2 ρ→1 ρ→1 ∂ρ ∂ρ e C1 | | 3 1 and consequently, φ is a extension of φ to z < 2 .Let0<ε< 4 be fixed and let ψ be a C∞ function on C such that 0 ≤ ψ ≤ 1, ψ(z)=1for|z| < 1+ε, | | 3 − ˜ e C1 and ψ(z)=0for z > 2 ε. We consider the function φ = ψφ , which is a compactly supported function. We have |∇ ˜ |2 |∇ e |2 | e |2 φ(z) φ (z) φ (z) 2 dv(z) 2 dv(z)+ 2 dv(z)=I + II. C V6K (z) | | 3 V6K (z) | | 3 V6K (z) z < 2 z < 2 In order to estimate the integrals in both I and II, we split both integrals into two D ≤|| 3 pieces, one corresponding to , and the other corresponding to 1 z < 2 .In this last region, we will apply an appropriate change of variables to transform the integrals into integrals over a subset of D.Wefirstobservethatforany00 such that for any h in the Dirichlet space D, | V |2 | |2 | |2 R (z) ≤ (I + R)h(z) h(z) 4 dv(z) C 2 dv(z). D V6K (z) D VK (z)

Proof. Since U is holomorphic, |RU| ≤ |∇V|.Ifρ<1, and hρ(z)=h(ρz), we apply Lemmas 3.3 and 3.4 to Re hρ and Im hρ, and we obtain that | V |2 |∇ |2 hρ(z)R (z) hρ(z) 4 dv(z) 2 dv(z). D V6K (z) D V2K (z)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BILINEAR FORMS ON THE DIRICHLET SPACE 2439

Next we have that VK (z) V2K (z) and consequently, by Fatou’s lemma, |∇V |2 |∇V |2 2 (z) 2 (z) |h(z)| dv(z) ≤ lim inf |hρ(z)| dv(z) V (z)4 ρ→1 V (z)4 D 6K D 6K |∇h (z)|2 |∇h(z)|2 lim inf ρ dv(z) dv(z). → 2 2 ρ 1 D V2K (z) D VK (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, |∇h(z)|2(1 −|z|2)−1dv(z) |Rh(z)|2(1 −|z|2)−1dv(z). Dα(ζ) Dβ (ζ)

V∗ 2 Consequently, multiplying by 1/( K ) (ζ), integrating and applying Lemma 2.7 to V∗ 2 the weight 1/( K ) (ζ) (which is also in the class A2), we easily obtain |∇ |2 | |2 h(z) Rh(z) 2 dv(z) 2 dv(z). D VK (z) D VK (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 | |2 | |2 Rh(z) (I + R)h(z) 2 dv(z) 2 dv(z). D VK (z) D VK (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

[1] D.R. Adams and L.I. Hedberg: Function spaces and potential theory. Grundlehren der Math- ematischen Wissenschaften 314. Springer-Verlag, Berlin, 1996. MR1411441 (97j:46024) [2] P. Ahern and J. Bruna: Maximal and area integral characterizations of Hardy-Sobolev spaces in the unit ball of Cn. Revista Matemática Iberoamericana 4 (1988), 123–152. MR1009122 (90h:32011) [3] P. Ahern and W. Cohn: Exceptional sets for Hardy Sobolev functions, p>1. Indiana Univ. Math. J. 38 (1989), 417–453. MR997390 (90m:32013) [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) [5] D. Békollé: Inégalitéà poids pour le projecteur de Bergman dans la boule unitédeCn. Studia Math. 71 (1981/82), no. 3, 305–323. MR667319 (83m:32004) [6] A.P. Calderon: Lebesgue spaces of differentiable functions and distributions. Proc Sympos. Pure Math. Vol. IV, 33–49, American Mathematical Society, Providence, R.I. MR0143037 (26:603) [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 functions, Monographs and Studies in Mathematics 23, Pitman, 1985. MR785568 (87j:46074) [10] V.G. Mazya and I.E. Verbitsky: The Schrödinger operator on the energy space: boundedness and compactness criteria. Acta Math. 188 (2002), 263–302. MR1947894 (2004b:35050)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2440 CARME CASCANTE AND JOAQUIN M. ORTEGA

[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 Matematica` Aplicada i Analisi,` Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain E-mail address: [email protected] Departmento Matematica` Aplicada i Analisi,` Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use