DIRECT AND REVERSE CARLESON MEASURES FOR H (b) SPACES. 1. Introduction

Transcription

1 DIRECT AND REVERSE CARLESON MEASURES FOR H (b) SPACES ALAIN BLANDIGNÈRES, EMMANUEL FRICAIN, FRÉDÉRIC GAUNARD, ANDREAS HARTMANN, AND WILLIAM T. ROSS Abstract. In this paper we discuss direct and reverse Carleson measures for the de Branges-Rovnyak spaces H (b).. Introduction Suppose H is a Hilbert space of analytic functions on the open unit disk D endowed with a norm H. If µ M + (D ), the positive finite Borel measures on the closed unit disk D, we say that µ is a Carleson measure for H if f µ f H for all f H and a reverse Carleson measure for H if f H f µ for all f H. Here µ is the L 2 (µ) norm and the notation f µ f H means there is a positive constant c µ so that f µ c µ f H for every f H (similarly for f H f µ ). One should notice that we need to address the technical difficulty of how to define f µ when µ places mass on part of the circle T := D since functions f H are a priori defined only on D. We will discuss this in the next section. Carleson measures have been discussed for many classical spaces of analytic functions in one and several variables and their wide-ranging applications in fields such as operator theory, interpolation, boundary behavior problems, and Bernstein inequalities demonstrate their importance. Recently, Lacey et al. [2] characterized the boundedness of the Cauchy transform which allows to answer the longstanding open question of characterizing the Carleson measures for the so-called model spaces. We will discuss how their results come into play in discussing 200 Mathematics Subject Classification. 30J05, 30H0, 46E22. Key words and phrases. de Branges-Rovnyak spaces, Carleson measures, reproducing kernel thesis, extreme point, corona pairs. This work is supported by ANR FRAB (ANR-09-BLAN ) and Labex CEMPI (ANR--LABX ).

2 2 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Carlesons measures for de Branges-Rovnyak spaces H (b) (defined below) which is one focus of this paper. On the other hand, reverse Carleson measures appeared more recently, though some early results go back to the 980s ([24, 25]), see also the recent paper [6]. A central paper in this direction is [22] where the authors were interested in composition operators. Another natural field where reverse, together with direct, Carleson measures are of interest concerns sampling problems, or more generally sampling measures. Of particular interest for us is the possibility of characterizing the rather implicitly defined norm in de Branges-Rovnyak spaces by an integral norm. In this paper we aim to accomplish three things. First, we continue the investigation begun in [4] which gives sufficient conditions for Carleson measures for H (b) spaces. Second, we continue an investigation of reverse Carleson measures for H (b), where b is non-extreme, begun in [5], where b was an inner function. Third, we relate our work to the recent paper of Lacey et al. [2] which examines the boundedness of the Cauchy transform operator. To put our work in some context, we review what is known for the classical Hardy space H 2 of the open unit disk. There are two main ways of describing Carleson measures for H 2. One is based on the H 2 - norm 2 of the reproducing kernels k λ (z) = ( λz) for H 2. The second one, a geometric description of Carleson (see [7]), is based on the ratio of µ(s(i))/m(i), where, for an open arc I in T, (.) S(I) := { z D : z/ z I, z m(i)/2 } denotes the Carleson window over I, and m is normalized Lebesgue measure on T. The well-known summary result is the following: Theorem.2. For µ M + (D ), the following are equivalent: () The inequality f µ f 2 holds for every f C(D ) H 2, where C(D ) are the complex valued continuous functions on D ; (2) The inequality k λ µ k λ 2 holds for every λ D; (3) The measure µ satisfies sup I µ(s(i)) m(i) < +, where the supremum is taken over all arcs I T.

3 DIRECT AND REVERSE CARLESON MEASURES 3 The implication (2) = () is known as the reproducing kernel thesis. Notice how the third statement of the theorem says that the Radon- Nikodym derivative of µ T with respect to m, is bounded on T and thus, via radial limits and Fatou s lemma, the integrals f µ, as well as the inequality f µ f 2, make sense for every f H 2 and not merely the continuous ones. For reverse Carleson measures there is the more recent result [8, 22]. Theorem.3. For µ M + (D ), the following are equivalent: () The inequality f 2 f µ holds for every f C(D ) H 2 ; (2) The inequality k λ 2 k λ µ holds for every λ D; (3) The measure µ satisfies inf I µ(s(i)) m(i) > 0, where the infimum is taken over all arcs I T. (4) The Radon-Nikodym derivative of µ T with respect to Lebesgue measure m is bounded below by a positive constant. Notice the appearance (again) of the reproducing kernel thesis (implication (2) = () of the theorem). Contrarily to direct embeddings, in the reverse situation, we cannot extend condition () to arbitrary H 2 - functions. Still, reproducing kernels are continuous up to the boundary so that condition (2) is perfectly meaningful. This will no longer be true in the subspaces we have in mind (for both direct and reverse Carleson measures), and which will require some additional conditions ensuring that at least kernels can be integrated against µ, see Definition 2. below. As stated earlier, we wish to discuss Carleson and reverse Carleson measures for the reproducing kernel Hilbert space H (b) of analytic functions on the open unit disk D whose reproducing kernel is given by kλ(z) b := b(λ)b(z), λ, z D. λz Here b belongs to H, the closed unit ball in H, and H is the Banach algebra of bounded analytic functions on D normed with the supremum norm. The space H (b) is often known as the de Branges-Rovnyak space and we will review the basics of this space in a moment. For now, note that if b <, then H (b) is just

4 4 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS the classical Hardy space H 2 with an equivalent norm while if b is an inner function, meaning b = almost everywhere on T = D, then H (b) is the classical model space (bh 2 ) = H 2 bh 2. For any b, the space H (b) is contractively contained in H 2. These spaces play an important role in operator theory since, via the compressed shift and the Sz.-Nagy-Foias and de Branges-Rovnyak theories, they serve as the model operator for a wide class of contraction operators on Hilbert spaces [28]. Acknowledgement: The authors would like to thank Dan Timotin for some valuable suggestions on the organization of this paper. 2. Main results As is often the case, the properties of H (b) spaces depend on whether or not b is an extreme point of H, i.e., log( b ) L (T, m) [4]. Since we have a complete description of the reverse Carleson measures for H (b) when b is non-extreme, we start with them. In order to give a precise definition of reverse Carleson measures, we need to finally address the technical difficulty of making sense of the integrals f µ when the measure µ has part of the unit circle T in its carrier. As mentioned earlier, it is not quite clear if a generic f H (b) can be defined on T in some reasonable way (perhaps via radial boundary values) µ-almost everywhere. In a way, we want to be as broad as possible as to not impose a too stringent condition on µ, and so we will only require the reverse inequality to hold on a dense set in H (b). This leads us to the following definition: Definition 2.. We say an f H (b) is µ-admissible if the radial limits of f exist µ-almost everywhere. We let H (b) µ denote the set of µ-admissible functions in H (b). Remark 2.2. Since the set of points in T where f has a radial limit is a Borel set [2, p. 23], the boundary function is a Borel function. With this definition in mind, if f H (b) µ, then defining f on the carrier of µ T via its radial boundary values, we see that f µ is well defined with a value in [0, + ]. Of course if µ is carried on D, i.e., µ(t) = 0, then H (b) µ = H (b). So Definition 2. only becomes meaningful when µ has part of the unit By a carrier of a measure µ M + (D ) we mean a Borel set C D for which µ(a C) = µ(a) for all Borel subsets A D.

5 DIRECT AND REVERSE CARLESON MEASURES 5 circle T in its carrier. Certainly for (normalized) Lebesgue measure m on T we know that H (b) m = H (b) (since H (b) H 2 ). However, there are often other µ, even ones with non-trivial singular parts on T with respect to m, for which H (b) = H (b) µ. The Clark measures associated with an inner function b have this property (see [5, 9]). If b is a µ-admissible function, then so are all finite linear combinations of reproducing kernels kλ b and thus, with this admissibility assumption on b, H (b) µ is a dense linear manifold in H (b). When b is a non-extreme point of H, then H (b) µ contains H (b) C(D ) which also turns out to be dense (see Section 3 below). This motivates our definition of reverse Carleson measure. Definition 2.3. For µ M + (D ) and b H we say that µ is a reverse Carleson measure for H (b) if H (b) µ is dense in H (b) and f b f µ for all f H (b) µ. In this definition, we allow the possibility for f µ to be infinite. We are now ready to state our main reverse Carleson result. Theorem 2.4. Let µ M + (D ) and let b be a non-extreme point of H and µ-admissible. Then the following assertions are equivalent: () The measure µ is a reverse Carleson mesure for H (b); (2) The inequality k b λ b k b λ µ holds for every λ D; (3) The measure ν defined by dν := ( b )dµ, satisfies inf I ν (S(I)) m(i) > 0; (4) If h = dµ T/dm, we have ess inf T ( b )h > 0. As mentioned earlier, when b < then H (b) = H 2 with equivalent norms and Theorem 2.4 corresponds to Theorem.3. Note that with (2) = () we again get the (reverse) reproducing kernel thesis [29, 32]. When b is an inner function, then H (b) is the classical model space (bh 2 ) and reverse Carleson measures for these spaces were discussed in our recent paper [5]. As it turns out there is no (reverse) reproducing kernel thesis in this setting [8]. For a general, possibly extreme b, we will have the following.

6 6 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Theorem 2.5. Suppose µ M + (D ) and b H is µ-admissible. If h = dµ T/dm and µ is a reverse Carleson measure for H (b) then (2.6) ( b 2 ) ( b 2 ) 2 h m-almost everywhere on T. When b is inner, the inequality in (2.6) is trivial, while in other cases, as we will see now, it yields important information: Corollary 2.7. Suppose µ M + (D ), b H is µ-admissible and not inner, h = dµ T/dm, and Z b := {ζ T : b(ζ) < }. If µ is a reverse Carleson measure for H (b) then h 0 and dm <. Z b b The above corollary says that any reverse Carleson measure for H (b), when b is not inner, must have a non-zero absolutely continuous component with respect to m. In particular, there cannot be sampling sequences when b is not inner (see Corollary 5.6). Notice how this is quite the dichotomy from the inner case where a reverse Carleson measure can be carried by D or even be singular with respect to m. We now define Carleson measures for H (b). Definition 2.8. A measure µ M + (D ) is a Carleson measure for H (b) if H (b) µ = H (b) and f µ f b for all f H (b). A result of Aleksandrov [2] shows that when b is inner then H (b) = (bh 2 ) contains a dense set of continuous functions. Furthermore, if f µ f b holds for (bh 2 ) C(D ) then every function in (bh 2 ) is µ-admissible and f µ f b for all f (bh 2 ). For a non-extreme b there is a unique outer function a with a(0) > 0 and such that a 2 + b 2 = m-almost everywhere on T (see Section 2). Here is one of our results concerning Carleson measures. Theorem 2.9. Let b be a rational and non-extreme point of H let µ M + (D ). Then the following assertions are equivalent: () The measure µ is a Carleson measure for H (b); (2) The measure a 2 dµ is a Carleson measure for H 2. and Observe that when b is rational then so is a [36, Remark 3.2] and so a 2 dµ is clearly defined even if µ T has a non-trivial singular part with

7 DIRECT AND REVERSE CARLESON MEASURES 7 respect to m (a point mass for example). Note that not every rational function in H is non-extreme (e.g., a finite Blaschke product). See Section 5 where we consider a more general result. When b is inner then, as mentioned earlier, H (b) = (bh 2 ) and Carleson measures for these spaces were discussed in [, 3,, 37, 38] (see also [20, 23, 30, 3]). Recall that Lacey et al [2] solved the longstanding problem of characterizing two-weight inequalities for Cauchy transforms. Let us take a moment to indicate how their results extend to H (b). Let σ be the Aleksandrov-Clark measure associated with b, that is the unique positive measure on T satisfying b(z) 2 b(z) = z 2 dσ(ζ), z D. 2 z ζ 2 T Let V b : L 2 (σ) H (b) be the operator defined by f(ζ) (V b f)(z) = ( b(z)) ζz dσ(ζ) = ( b(z))(c σf)(z), T where C σ is the Cauchy transform. It is known [35] that V b is a partial isometry from L 2 (σ) onto H (b) and ker V b = ker C σ = (H 2 (σ)). Here H 2 (σ) denotes the closure of polynomials in L 2 (σ). As a consequence, since every function f H (b) can be written as f = V b g for some g H 2 (σ), µ is a Carleson measure for H (b) if and only if V b g µ = f µ f b = V b g b = g σ, Setting ν b,µ := b 2 µ, we have V b g 2 µ = b 2 C σ g 2 dµ = C σ g 2 ν b,µ, D which yields the following: g H 2 (σ). Theorem 2.0. Let µ M + (D ), b a µ-admissible function in H, and set ν b,µ := b 2 µ. The following are equivalent: (i) The measure µ is a Carleson measure for H (b); (ii) The Cauchy transform C σ is a bounded map from L 2 (σ) into L 2 (D, ν b,µ ), where σ is the Clark measure associated with b; We refer the reader to [2, Theorem.7] for a description of the boundedness of the Cauchy transform operator C σ. We also examine, in the non-extreme case, when the L 2 (µ)-norm defines an equivalent norm of H (b), i.e., when each f H (b) is µ-admissible

8 8 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS and f b f µ. This does indeed occur but only under very stringent circumstances. If we were to require the stronger condition that µ is an isometric measure, i.e., f b = f µ for all f H (b), then this never occurs: Theorem 2.. If b is non-constant and a non-extreme point of H, then there are no positive isometric measures for H (b). When b is inner, there are plenty of isometric measures [, 5]. 3. Preliminary facts about H (b) For a detailed treatment of de Branges-Rovnyak spaces, we refer the reader to Sarason s book [35] which contains all the material presented in this section. We merely set the notation and remind the reader of some standard facts we will use later. For φ L := L (T, m), define the Toeplitz operator on the Hardy space H 2 by T φ f = P + (φf), f H 2, where P + is the orthogonal projection (the Riesz projection) of L 2 := L 2 (T, m) onto H 2. Note, as is standard, how we regard H 2 as both a Hilbert space of analytic functions on D and, via radial boundary values and Fourier coefficients, a closed subspace of L 2 [5, 7]. We will use f, g 2 := f(ζ)g(ζ)dm(ζ) T for the inner product on H 2 (or L 2 ) and f 2 to denote the norm. Also note that when φ H, the bounded analytic functions on D, we have T φ f = φf for all f H 2. For b H the de Branges-Rovnyak space H (b) is defined to be endowed with the inner product H (b) := (I T b T b ) /2 H 2, (I T b T b ) /2 f, (I T b T b ) /2 g b := f, g 2, for f, g ker((i T b T b ) /2 ). That is to say, H (b) is normed to make (I T b T b ) /2 a partial isometry of H 2 onto H (b). When b <, I T b T b is an isomorphism on H 2 and thus H (b) = H 2 with an equivalent norm. On the other extreme, when b is an inner function then T b T b is the orthogonal projection of H 2 onto bh 2 and thus H (b) is (bh 2 ) = H 2 bh 2 with the standard H 2 norm.

9 DIRECT AND REVERSE CARLESON MEASURES 9 The space H (b) is a reproducing kernel space with kernel kλ(z) b := b(λ)b(z), λ, z D, λz i.e., f(λ) = f, kλ b b for all λ D and f H (b). We point out that if k λ (z) := ( λz) is the standard reproducing kernel for H 2, then kλ b = (I T bt b )k λ. Observe the notation here: kλ b is the reproducing kernel for H (b) while k λ is the reproducing kernel for H 2. We will now assume for the main part of the paper that b is a nonextreme point of the unit ball of H, which, by the Arens Buck Carleson Hoffman Royden [4] theorem, is equivalent to the condition log( b ) L = L (T, m). To abbreviate, we will simply say b is non-extreme. In this case, one can let a be the outer function whose modulus is equal to b 2 almost everywhere on T and a(0) > 0. Then (3.) a(ζ) 2 + b(ζ) 2 = m a.e. ζ T, and we call a pair (a, b) satisfying the above equality a Pythagorean pair and the outer function a the Pythagorean mate for b. For φ L, let M (φ) := T φ H 2 endowed with the norm which makes T φ a partial isometry from H 2 onto T φ H 2. Observe that when a H is outer, then T a and T a are one-to-one and hence T a f M (a) = T a f M (a) = f 2, f H 2. When b is non-extreme then M (a) = ah 2 is contractively contained in M (a) which is, in turn, contractively contained in H (b). Moreover, M (a) is dense in H (b) and the linear span of the reproducing kernels k λ (for H 2 ), λ D, is contained and dense in M (a) and thus it is also dense in H (b). In particular, we see that H (b) C(D ) is dense in H (b). It is also known, when b is non-extreme, that for every f H (b), we have T bf M (a) and (3.2) f 2 b = f g 2 2, where g is defined by (3.3) T b f = T a g.

10 0 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Note that g is unique since, as discussed earlier, T a is one-to-one due to the fact that a is outer. At least when (b/a)f L 2, it can be checked that (3.4) g = T b/a f. We say that (a, b) forms a corona pair if inf{ a(z) + b(z) : z D} > 0. Still under the hypothesis that b is non-extreme, we have that M (a) = H (b) (with equivalent norms) if and only if (a, b) forms a corona pair. We also have M (a) = H (b) (with equivalent norms) if and only if (a, b) forms a corona pair and T a/a is invertible. Recall [27, Theorem B4.3.] that T a/ā is invertible if and only if a 2 satisfies the Muckenhoupt (A 2 ) condition, i.e., ( ) ( ) (3.5) sup a 2 dm a 2 dm <, I m(i) m(i) I where I runs over all subarcs of T. For shorthand we will often write a 2 (A 2 ) when a 2 satisfies (3.5). The (A 2 ) condition is equivalent to the boundedness of the Riesz projection P + from L 2 ( a 2 dm) to itself (or from L 2 ( a 2 dm) to itself). Note that if (a, b) is a Pythagorean pair, the µ-admissibility of one function does not imply that of the second one. Indeed pick µ = δ, the point mass at, and let a 0 be an outer function bounded by which has no radial limit at. Multiply a 0 by the atomic inner function I(z) = exp((z + )/(z )). Then a = a 0 I has a radial limit 0 at. Now a and a 0 have the same Pythagorean mates e iθ b (θ R). Then either b has radial limit at and a 0 does not or b does not have a radial limit at but a does. I 4. A first observation about reverse Carleson measures When b is non-extreme there is an interesting relationship between reverse Carleson measures and the Pythagorean pair (a, b). Recall from the previous section that since b is non-extreme, the reproducing kernels k λ (z) = ( λz) for H 2 belong to H (b). Proposition 4.. If b H is non-extreme and µ M + (D ) satisfies k λ b k λ µ for all λ D, then b/a H 2.

11 DIRECT AND REVERSE CARLESON MEASURES Proof. Since T bk λ = b(λ)k λ = (b(λ)/a(λ))tāk λ, we get from (3.2) ) (4.2) k λ 2 b = k λ b(λ) 2 a(λ) k λ = ( + b(λ) 2 a(λ) 2 λ. 2 Hence there is a constant C > 0 such that b(λ) 2 D a(λ) C λ 2 dµ(z), λ D. 2 λz 2 Setting λ = re it, integrating both sides of the previous inequality over t (0, 2π), and using Fubini s theorem, we obtain 2π b(re it ) 2 ( 2π ) r 2 dt dt C dµ(z). 2π a(re it ) 2 rze it 2 2π 0 Using basic properties of the Poisson kernel we get 2π 0 D r 2 rze it 2 dt 2π = for every z D, which yields 2π 2π for every r (0, ). Hence b/a H r2, r 2 z 2 b(re it ) 2 a(re it ) 2 dt Cµ(D ) < We can connect this proposition to some well-known equivalences [34]. Theorem 4.3 (Sarason). Let (a, b) be a Pythagorean pair. Then the following assertions are equivalent: () The function b/a belongs to H 2 ; (2) The space H is contained in H (b); (3) We have sup n N z n b < ; (4) The function ( b ) belongs to L. Remark 4.4. If b/a satisfies the stronger condition b/a H, then b < and so H (b) = H 2 (with equivalent norms). To see this, write b = ah, for some h H. It follows that = a 2 + b 2 = a 2 ( + h 2) a.e. on T. From here we get /a L and so b <. Not every H (b) space admits a reverse Carleson measure. Indeed, we have the following:

12 2 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Corollary 4.5. Let b H be non-extreme. Then H (b) admits a reverse Carleson measure if and only if ( b ) L. Thus there are H (b) spaces with non-extreme b which do not admit reverse Carleson measures. Proof. Suppose H (b) admits a reverse Carleson measure. By Proposition 4., along with Theorem 4.3, we see that ( b ) L. The converse will follow from Theorem 5. below (see Remark 5.2). For a specific example of when H (b) admits no reverse Carleson measures set b(z) = ( z)/2 and note that /( b ) L. 5. The Main reverse Carleson measure result For µ M + (D ), recall that b is µ-admissible if the radial limits of b exist µ-almost everywhere on T. Also recall that when b is µ-admissible then H (b) µ, the set of all µ-admissible functions in H (b), is a dense linear manifold in H (b). Let us restate our main reverse Carleson measure result from the introduction (Theorem 2.4) for convenience of the reader. Theorem 5.. Let µ M + (D ), b H be non-extreme and µ- admissible. Then the following are equivalent: () The measure µ is a reverse Carleson measure for H (b); (2) The inequality k b λ b k b λ µ holds for every λ D; (3) The measure ν defined by dν := ( b 2 )dµ satisfies the condition ν (S(I)) (5.2) inf > 0; I m(i) (4) If h = dµ T/dm, we have ess inf T ( b 2 )h > 0. Remark 5.3. () Notice that since (2) = (), the reverse reproducing kernel thesis holds when b is non-extreme. This is not always the case when b is extreme [8]. (2) As in H 2, the part of the measure guaranteeing reverse Carleson embeddings for H (b) is supported on T. The proof of Theorem 5. will require the following technical lemma from harmonic analysis.

13 DIRECT AND REVERSE CARLESON MEASURES 3 Lemma 5.4. Let q H. Then for almost every ζ T, (5.5) lim q(rξ) 2 r2 r ξ rζ dm(ξ) = 2 q(ζ) 2. T Proof. Suppose ζ T is a Lebesgue point of q 2 where q admits a radial limit l. We can assume l = 0. Let u be the harmonic function on D whose (almost everywhere defined) radial limits u(ξ), ξ T, satisfy u(ξ) = q(ξ) 2 a.e. Then, by the fact that q 2 is subharmonic, we have q 2 u on D and it follows that 0 T q(rξ) 2 r2 dm(ξ) ξ rζ 2 By our assumptions on ζ, the above implies T u(rξ) r2 ξ rζ 2 dm(ξ) = u(r2 ζ). (5.6) 0 lim q(rξ) r T 2 r2 ξ rζ dm(ξ) 2 q(ζ) 2 = l 2 = 0. Clearly we can switch to a regular limit which completes the proof. Remark 5.7. It follows from Lemma 5.4 and basic facts about the Poisson kernel that, for any arc I, lim q(rξ) 2 r2 r ξ rζ dm(ξ) 2 I is equal almost everywhere to q(ζ) 2 when ζ is in the interior of I and zero when ζ is not in the closure of I. Proof of Theorem 5.. We will prove: () (2) (4) (3) and (4) (). () implies (2) is clear since b is µ-admissible and so k b λ H (b) µ for every λ D. (4) (3) is exactly the same as in Theorem.3 (see [8] for the proof). For (4) () let δ := ess inf T ( b 2 )h. To test the reverse Carleson condition, we just need to test it on f H (b) µ with f µ <. For such f we have, with a being the Pythagorean mate for b, f 2 a 2 dm δ f 2 h dm = δ f 2 dµ <. T T Thus f/a L 2. But since a is outer, this yields (via a standard fact from Hardy spaces) that f/a H 2 and thus f = af/a M (a). Now T

14 4 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS use the fact that M (a) is contractively contained in H (b) to get which is the desired inequality. f b f M (a) = f/a 2 δ /2 f µ We now prove (2) (4). If kλ b b kλ b µ for all λ D then b(λ) 2 2 b(λ)b(z) dµ(z). λ 2 D λz This says that (5.8) b(λ) 2 where + T D λ 2 λz 2 b(λ)b(z) 2 dη(z) λ 2 T λz 2 b(λ)b(z) 2 h(z)dm(z) λ 2 + λz 2 b(λ)b(z) 2 dσ(z), dµ = dη + hdm + dσ, η = µ D, h = dµ T/dm, and σ is the singular part of µ T. Note that the three integrals on the right-hand side of (5.8) are well defined since we are assuming that b is a µ-admissible, bounded analytic function. Let E be the set of points ζ T which satisfies the three conditions (i) ζ is a Lebesgue point of h; (ii) b has a radial limit at ζ; (iii) (Dσ)(ζ) = 0; where Dσ is the symmetric derivative of σ (which is zero m-almost everywhere). Notice that E is a set of full Lebesgue measure in T. Let I = (e iθ, e iθ 2 ) be a sub-arc of T containing ζ E and define (5.9) S(I, y) := {z S(I) : z y}. Integrating the left-hand side of (5.8) over S(I, y) and dividing by y we get ( θ2 ) ( b(re it ) 2 )dt rdr. y y θ After an application of Fubini s theorem, followed by an application of the dominated convergence theorem, this quantity goes to θ2 ( b(e it ) 2 )dt as y 0. θ

15 DIRECT AND REVERSE CARLESON MEASURES 5 Now integrate the first integral on the right-hand side of (5.8) over S(I, y) and divide by y to get (again after applying Fubini s theorem) ( θ2 ) r 2 (5.0) D y y θ re it z 2 b(reit )b(z) 2 dt rdrdη(z). Note that the inner double integral, i.e., y y ( θ2 r 2 θ re it z 2 b(reit )b(z) 2 dt is bounded above by a constant times ( θ ) r 2 y θ re it z dt rdr 2 y ) rdr which approaches χ I as y 0. Thus the integral in (5.0) is bounded above by a quantity which approaches (5.) χ I dη as y 0, which is equal to zero since I D =. D Now integrate the second integral on the right-hand side of (5.8) over S(I, y) and divide by y to get (after applying Fubini s theorem) ( θ ) r 2 T y y θ e it rζ 2 b(reit )b(ζ) 2 dt rdrh(ζ)dm(ζ). Apply Lemma 5.4 and Remark 5.7 to the inner integral θ2 r 2 θ e it rζ 2 b(reit )b(ζ) 2 dt to see that this quantity approaches ( b(ζ) 2 ) 2 χ I as r. Thus the second integral on the right-hand side of (5.8) (after integrating over S(I, y) and dividing by y) approaches ( b 2 ) 2 hdm, as y 0. I Now use the exact same proof used to get (5.) to show that the third integral on the right-hand side of (5.8) (after integrating over S(I, y) and dividing by y) approaches dσ as y 0. I Combining our results we get ( b 2 )dm ( b 2 ) 2 hdm + I Remembering that Dσ is zero on E, we get the required result. I I dσ.

16 6 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Remark 5.2. () The proof can be modified to show that the H 2 kernels k λ (which are contained in H (b) since b is non-extreme see Section 3) can be used to test the reverse embedding. More precisely, if k λ b k λ µ for every λ in D, then µ is a reverse Carleson measure. Since k λ is simpler than kλ b, this could potentially provide an easier test for reverse Carleson measures. (2) Theorem 5. can be used to complete the proof of the converse of Corollary 4.5. Indeed if ( b ) L then the measure dµ := ( b ) dm is finite, b is admissible with respect to this measure, and µ is a reverse Carleson measure for H (b) since ess inf T ( b 2 )( b ) > 0. For general b H, the proof of (2) (4) in Theorem 5. actually shows the following. Theorem 5.3. Suppose µ M + (D ), b H is µ-admissible, and h = dµ T/dm. If µ is a reverse Carleson measure for H (b) then (5.4) ( b 2 ) ( b 2 ) 2 h. almost everywhere on T. When b is inner then (5.4), though true, yields no information. When b is non-extreme then the condition that m({ b = }) = 0 implies that condition (5.4) is equivalent to (4) in Theorem 5.. We have the following: Corollary 5.5. Suppose µ M + (D ), b H is not inner and µ- admissible. Let h = dµ T/dm and Z b := {ζ T : b(ζ) < }. Assume that µ is a reverse Carleson measure for H (b), then h 0 and dm < +. Z b b Moreover, if m(z b ) =, then b is non-extreme. Proof. By Theorem 5.3, the inequality (5.4) holds and since b is not inner, this inequality implies that h 0. On Z b we obtain from (5.4) that ( b 2 )h, that is ( b ) h a.e. on Z b. Since h L (T), we see that Z b ( b ) dm is finite. If furthermore m(z b ) =, the integrability of ( b ) implies that of log( b ) and so b is nonextreme.

17 DIRECT AND REVERSE CARLESON MEASURES 7 Our results have an application to sampling sequences for H (b). Recall that if H is a reproducing kernel Hilbert space on D with kernel k H λ, then a sequence {λ n } n D is called a sampling sequence for H if f 2 H n= k H λ n 2 H f(λ n) 2, f H. Corollary 5.6. Let b H. If H (b) admits a sampling sequence, then necessarily b is an inner function. Proof. Assume {λ n } n is a sampling sequence for H (b). Then µ := n= kλ b n 2 b δ λn is a reverse Carleson measure for H (b). But since dµ T/dm 0, Corollary 5.5 implies that b is an inner function. This Corollary generalizes a result from [8]. Note also that the proof given above shows that H (b) does not have an orthogonal basis of reproducing kernels of the form kζ b, where ζ T, if b is not inner. This result was shown in [6] using a different proof. Example 5.7. Let b be the outer function whose modulus satisfies b(e iθ ) = exp( θ 2 ). Then T log( b ) dm 2π 0 θ 2 dθ = and so b is extreme. Moreover, Z b = T \ {}. In particular, m(z b ) = and by Corollary 5.5, H (b) admits no reverse Carleson measures. Conspicuously missing from this discussion is the case where b is extreme and not inner and for which dm <. Z b b An example of this would be the outer function b whose modulus is on T {Iz > 0} and /2 on T {Iz < 0}. Do such H (b) spaces have reverse Carleson measures?

18 8 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS 6. An analogous condition for direct embeddings In this section, we assume b is non-extreme and a is its Pythagorean mate. We also recall the definition of Carleson measure from the introduction: H (b) µ = H (b) and f µ f b for every f H (b). We remind the reader that M (a) = ah 2 is contractively contained in H (b), i.e., for every g H 2, (6.) ag b ag M (a) = g 2. In particular, a H (b) and thus, if µ is a Carleson measure for H (b), then necessarily a is µ-admissible. Proposition 6.2. Let b H be non-extreme and let µ M + (D ) be a Carleson measure for H (b). Then dν = a 2 dµ is a Carleson measure for H 2. Proof. For λ D and k λ the reproducing kernel for H 2, apply the direct embedding inequality to ak λ and use (6.) to obtain k λ 2 = ak λ M (a) ak λ b ak λ µ = k λ ν, λ D, which implies, by the reproducing kernel thesis for H 2, that ν is a Carleson measure for H 2. Remark 6.3. We will see in a moment that without additional assumptions on a and b, the converse is not always true. For our next set of results we need some additional facts concerning H (b) spaces. For α T let σ α be the Aleksandrov Clark measure [9, 0, 33] associated with ᾱb, i.e., σ α is (via a classical theorem of Herglotz) the unique positive measure on T satisfying (6.4) b(z) 2 ᾱb(z) 2 = T z 2 ζz 2 dσ α(ζ), z D. According to [35, IV-0], since b is non-extreme, σ α m for m-almost every α T. Now let F α := a ᾱb and note that F α belongs to H 2 and is outer. From [35, IX-4] we know that M (a) = H (b) (with equivalent norms) if and only if there is an α T such that σ α m and F α 2 (A 2 ) (recall (3.5)). The following theorem may seem overly technical at first glance but it will have a very useful corollary (See Corollary 6.7 below).

19 DIRECT AND REVERSE CARLESON MEASURES 9 Theorem 6.5. Let (a, b) be a Pythagorean pair, µ M + (D ), α T such that σ α m. Assume there exists a polynomial p having all of its roots on T and an f H 2 satisfying the conditions f 2 (A 2 ) and F α = pf. Then the following assertions are equivalent: () The measure µ is a Carleson measure for H (b); (2) The function a is µ-admissible and the measure a 2 dµ is a Carleson measure for H 2. Proof. The implication () (2) has already been proved in a general setting. Let us now focus on the reverse implication and assume that dν := a 2 dµ is a Carleson measure for H 2. Write s p(z) = (z ζ i ) m i, i= where, by hypothesis, ζ i T. Let N = m + m m s denote the degree of p. According to [35, X-8], we know that M (a) is closed in H (b) with co-dimension N. Note that all functions in M (a) are µ-admissible (because ν is a Carleson measure for H 2 ). If N = 0 then F α 2 (A 2 ) and we know that M (a) = H (b) with equivalent norms. Then for every f = ag H (b), we have f µ = g ν g 2 = ag M (a) f b, which proves the desired embedding. Now assume N and let us first show that H (b) can be written as (6.6) H (b) = M (a) P N, where the sum in the above decomposition is direct (not necessarily orthogonal). First note that since b is non-extreme, the polynomials belong to H (b). Now let q P N M (a). That means that the polynomial q can be written as q = ag for some g H 2. But then, since pf = a αb, we see that the rational function q = ( ᾱb)fg p belongs to H. This is clearly possible if and only if the poles of q/p are outside D. In particular, we see that the polynomial q should have a zero of order at least m i at each point ζ i. Since the degree of q is less or equal to N, this necessary implies that q = 0. Hence the sum

20 20 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS M (a) P N is direct. But since dim P N = N = codim M (a), we obtain (6.6). In particular, the angle between the subspaces M (a) and P N is strictly positive which means that f b ag b + p b, for every f = ag + p H (b) where ag M (a) and p P N. Moreover, since M (a) is a closed subspace of H (b), contractively embedded ( ag b ag M (a), g H 2 ), the open mapping theorem shows that ag M (a) ag b, g H 2. Note that H (b) µ = H (b), and since ν is a Carleson measure for H 2, we have, for ag M (a), ag µ = g ν g 2 = ag M (a) ag b. On the other hand, if p(z) = N n=0 a nz n P N, we see, since µ is a finite measure, that ( N N N ) /2 p µ a n z n µ a n a n 2 = p 2 p b. n=0 n=0 We conclude that for f = ag + p, where ag M (a) and p P N, we have n=0 f µ = ag + p µ ag µ + p µ ag b + p b f b. A nice application of Theorem 6.5 is the following corollary (stated earlier as Theorem 2.9). Note that if b is a rational function then so is a [36, Remark 3.2] so there is no need to impose any µ admissibility conditions. Corollary 6.7. Let b be rational and non-extreme and µ M + (D ). Then the following assertions are equivalent: () The measure µ is a Carleson measure for H (b); (2) The measure a 2 dµ is a Carleson measure for H 2. Proof. According to Theorem 6.5, it is sufficient to prove that there exists an α T such that σ α from (6.4) satisfies σ α m, a polynomial p having all of its roots on T, and a function f H 2 with f 2 (A 2 ) such that F α = pf. We first observe that the associated outer function a is also a rational function (see [36, Remark 3.2]). Write a = q/r where q and r are two

21 DIRECT AND REVERSE CARLESON MEASURES 2 polynomials with GCD(q, r) =. Then, necessarily, r(z) 0 for every z D and let us denote by ζ i, i N, the zeros of q on T. Note that since these zeros are the same as those of a we actually have {z T : b(z) = } = {ζ,..., ζ N }. Now choose α T \ {b(ζ ),..., b(ζ N )} such that σ α m (which is always possible because σ α m for m-almost every α T). Moreover, according to the choice of α, the function ᾱb, which is continuous on D, cannot vanish and hence (6.8) inf r(z)( ᾱb(z)) > 0. z D It remains to factor the polynomial q as q q 2 where q has all of its roots on T and q 2 has all of its roots outside D. Then F α = a ᾱb = q q 2 r( ᾱb) = q f, where f = q 2 r( ᾱb). In view of (6.8), we easily see that f and /f are continuous on D which implies f 2 (A 2 ). The proof is completed by an application of Theorem 6.5. Remark 6.9. As a byproduct of our proof of Corollary 6.7 and Theorem 6.5, we see that if b is rational and non-extreme and if ζ,..., ζ n are the zeros of a on T, listed according to multiplicity, then ( n ) H (b) = (z ζ j ) H 2 P n. j= This decomposition already appears in [3, Lemma 4.3] but their argument is based on a difficult result of Ball and Kriete which gives a condition as to when one H (b)-space is contained in another one. Moreover, if we gather [3, Theorem 4..] and Corollary 6.7, then we recover a result of [7] concerning the characterization of direct Carleson measures for a Dirichlet-type space associated with a finite sum of Dirac measures. We already mentioned in Section 3 that if (a, b) is a corona pair and if T a/a is invertible then H (b) = M (a). We also pointed out that this is equivalent to the fact that there exists α T with σ α (from (6.4)) satisfies σ α m and F α 2 (A 2 ). In particular, applying

22 22 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Theorem 6.5 to this situation (or by direct inspection) immediately gives the following result. Corollary 6.0. Suppose that (a, b) forms a corona pair and T a/a is invertible. If µ M + (D ), then the following assertions are equivalent: () The measure µ is a Carleson measure for H (b); (2) The function a is µ-admissible and the measure a 2 dµ is a Carleson measure for H 2. Remark 6.. A sufficient condition for direct Carleson measures for H (b) is given in [4]. More precisely, for ε (0, ), we let Ω(b, ε) denote the sub-level set Ω(b, ε) := {z D : b(z) < ε}, and σ(b) := {ζ T : lim inf z ζ b(z) < } denote the boundary spectrum of b and we set Ω(b, ε) = Ω(b, ε) σ(b). It is known [4, Theorem 6.] that if µ satisfies µ(s(i)) I, (recall the definition of the Carleson window S(I) from (.)) for any arc I such that S(I) Ω(b, ε), then f µ f b, f H (b). Note that our results, in some sense, complete the picture because one can, using Corollary 6.7, produce an example of a Carleson measure for H (b) where the above criterion cannot be applied. Indeed let b(z) = (+z)/2. Then a(z) = ( z)/2 and σ(b) = T\{}. Thus checking the Carleson condition for µ on Carleson squares which intersect Ω(b, ε) is, in this case, equivalent to saying that µ is a Carleson measure for H 2 at least around the point ζ =. But now it is easy to construct an example of a measure µ which is Carleson for H (b) but not for H 2. For instance, we can consider the measure µ on the interval (0, ) defined by dµ(t) = ( t) β dt, for β (0, 2]. According to Corollary 6.7, µ is a Carleson measure for H (b) but cannot be a Carleson measure for H 2 because if we consider the arc I ϑ := (e iϑ, e iϑ ), ϑ (0, π/2), we have µ(s(i ϑ )) = ϑ/2π dt ( t) = (ϑ/2π) β, β β

23 and thus DIRECT AND REVERSE CARLESON MEASURES 23 µ(s(i ϑ )) sup ϑ>0 I ϑ =. However, it is important to note here that our results focus on the case where b is non-extreme whereas in [4], there are no such assumptions on b. Remark 6.2. In [4], it is shown that for a particular class of functions b, the reproducing kernel thesis is true. More precisely, let b H and assume that there exists ε (0, ) such that Ω(b, ε) is connected and its closure contains the spectrum σ(b). If k b λ µ k b λ b holds for every λ D, then f µ f b for every f H (b). However, in [26], F. Nazarov and A. Vol berg showed that this is no longer true in the general case (their counterexample corresponds to the case where b is an inner function). According to the results of this paper, it would be natural to conjecture that the reproducing kernel thesis is true in the non-extreme case. However, we currently do not know how to prove this. 7. Examples We would like to discuss the necessity of the two hypotheses appearing in Corollary 6.0. To do so, we will construct two examples where either of the two conditions (7.) (a, b) is a corona pair (7.2) T a/a is invertible are violated and yet on may construct a measure µ M + (D ) such that a 2 dµ is Careson measure for H 2 and for which there is no Carleson embedding H (b) L 2 (µ). Let us start with condition (7.). Example 7.3. There is a Pythagorean pair (a, b) and a µ M + (D ) such that (a, b) is not a corona pair, T a/a is invertible, a 2 dµ is a Carleson measure for H 2, yet µ is not a Carleson measure for H (b). To see this, let a(z) := ( ) z α 2 for α (0, /2), guaranteeing that a 2 (A 2 ) (equivalently T a/a is invertible). Clearly a 2 is a bounded, logintegrable function, and so that there is an outer function b 0 H such that a 2 + b 0 2 = a.e. on T. Symmetrically, b 0 is also non-extreme, and its Pythagorean mate is a. Now consider the Blaschke product

24 24 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS B = B Λ whose zeros are Λ = { /2 n } n, and set b = Bb 0. Then (a, b) is not a corona pair since a(λ n ) + b(λ n ) = a(λ n ) 0, n. Now consider the measure µ on T defined by dµ(z) = z β dm(z) for some 0 < β 2α. Then a 2 dµ = c 2 z 2α β dm has bounded Radon-Nikodym derivative and is therefore a Carleson measure for H 2. To show that µ is not a Carleson measure for H (b), we now estimate the L 2 (µ)-norms of the normalized reproducing kernels κ λn (z) = k λ n (z) λn = 2 k λn b λ n z, z D. Note in the above that b(λ n ) = 0 and so k b λ n (z) = b(λ n)b(z) λ n z Clearly κ λn b =. However, κ λn 2 λ n 2 dm(ζ) L 2 (µ) = ζ λ n 2 ζ β 2 n T 2 n /2 n = λ n z = k λ n (z). π π /2 n e it λ n 2 t β dt 2 n 22n 2 nβ, n. Hence µ is not a Carleson measure for H (b). e it λ n 2 t dt β /2 n The following result discusses the necessity of condition (7.2). /2 n t β dt Theorem 7.4. Let (a, b) be a corona pair such that a 2 L and let dµ = a 2 dm. Then µ is a Carleson measure for H (b) if and only if a 2 (A 2 ). Proof. The following equivalences are quite obvious: M (a) L 2 (µ) f µ f M (a), f M (a), T a g µ g 2, g H 2, T a : H 2 L 2 (µ) is bounded T a : L 2 L 2 (µ) is bounded.

25 DIRECT AND REVERSE CARLESON MEASURES 25 Let j : L 2 L 2 (µ) be the onto isometry j(f) = af and observe that the following diagram commutes: L 2 j T a Hence L 2 (µ) P + L 2 (µ). M (a) L 2 (µ) P + : L 2 (µ) L 2 (µ) is bounded a 2 (A 2 ). Remembering that since (a, b) is a corona pair, then H (b) = M (a) with equivalent norms which completes the proof. Example 7.5. Based on Theorem 7.4, it is possible to construct an explicit example of a corona pair (a, b) so that dµ = a 2 dm is not a Carleson measure for H (b) whereas a 2 dµ = dm is naturally a Carleson measure for H 2. In view of Corollary 6.0, condition (7.2) will not be satisfied here. The idea is the following. Pick a decreasing sequence {β n } n (0, /2) and with n log β 2 2n n <. For n, introduce the intervals [ I n := 2 + 2n+ 2, 3n 2 ) [, J 2n 2 3n n := 2 + 2n 2, 3n 2 ) 2n 2 3n and define a symmetric function u (u(e it ) = u(e it )) on these intervals by { /2 if t u(e it Jn, ) = β n if t I n. Connect smoothly and monotonically the values /2 and β n between these intervals and set u(e it ) = /2 on the remaining part of the circle. By construction, u is a bounded log-integrable function. Let a be the outer function with a = u almost everywhere, which in view of the smoothness of u can be assumed continuous on the whole closed unit disk except at, and let b its Pythagorean mate. Since a is small only near, where b is large, the pair (a, b) is a corona pair. Let us check that a 2 is not (A 2 ). Consider the intervals. [ K n := 2 + 2n+ 2, 3n 2 ) I 2n 2 3n n J n, the length of which is comparable to 2 2n.

26 26 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS On K n we have K n a(e it ) 2 dt 2 2n K n J n u(e it )dt (the contribution of u on I n is neglectible and J n 2 2n ). On the other hand dt K n K n a(e it 22n ) 2 I n u(e it ) dt β n (the contribution of /u on J n is neglectible and I n 2 2n ). Hence ( ) ( ) a(e it ) 2 dt K n K n K n K n a(e it ) dt, n, 2 β n which proves the claim. According to Theorem 7.4 dµ = a 2 dm is not a Carleson measure for H (b). To finish this section we comment further on the more general situation dµ = hdm when h is not necessarily equal to a 2. Then the direct embedding result is connected with the so-called two-weight estimates mentioned earlier. Indeed, with a similar argument as in Theorem 7.4, we can show that when (a, b) is a corona pair such that a 2 L and dν = a 2 dµ for some absolutely continuous measure dµ = hdm on T, then, as in the proof of Theorem 7.4, M (a) L 2 (µ) T a : L 2 L 2 (µ) is bounded. As was done earlier, let us consider the corresponding commutative diagram. To do this set dγ = a 2 dm and j : L 2 L 2 (γ) be the onto-isometry j(f) := af. Then the following diagram commutes: L 2 j T a L 2 (γ) P + L 2 (µ) Hence, recalling that dµ = h dm, M (a) embeds into L 2 (µ) if and only if ) P + : L ( a 2 2 dm L 2 (h dm) is bounded. Using one more time that M (a) = H (b) (with equivalent norms), we get:

27 DIRECT AND REVERSE CARLESON MEASURES 27 Theorem 7.6. Let us suppose that (a, b) forms a corona pair such that a 2 L. Let dµ = h dm with h L. Then ) µ is a Carleson measure for H (b) if and only if P + : L ( a 2 2 dm L 2 (h dm) is bounded We again refer the reader to [2] for a characterization of the two-weight boundedness of P + as required in the theorem. 8. Norm equivalence and isometric embeddings This last section is devoted to a discussion of equivalent norms on H (b) and isometric embeddings of H (b) in L 2 (µ). It will turn out that the L 2 (µ) norm is equivalent to the H (b)-norm only when H (b) = M (a). We begin with an observation. If we are to impose the condition that the L 2 (µ) norm is equivalent to the H (b) norm then we require that H (b) µ = H (b) and that f µ f b for all f H (b). When b is non-extreme then Proposition 4. and Theorem 4.3 imply that H H (b). This means that if every function in H (b) has a radial limit µ-almost everywhere on T then every function in H must also have this property. By a classical result of Lusin [2, p. 24] (given any closed subset of T with Lebesgue measure zero there is an f H which does not have radial limits on this set), we see that the singular part of µ T, with respect to m, is zero. Proposition 8.. Let b H be a non-extreme point, µ M + (D ), and h = dµ T /dm. Assume that H (b) µ = H (b) and f µ f b, f H (b). Then (a, b) is a corona pair and a 2 (A 2 ). In other words, it is necessary that H (b) = M (a). Proof. First note that by our above discussion we have dµ T = hdm. Second, according to Theorem 5., we have (8.2) δ := ess inf T a 2 h > 0. Third, using (4.2), there exists a constant c > 0 such that c D λz 2 dµ(z) a(λ) 2 + b(λ) 2 a(λ) 2 ( λ 2 ) for every λ D. Hence with (8.2) we get 2 λ 2 cδ a(ζ) λζ dm(ζ) a(λ) 2 + b(λ) 2. 2 a(λ) 2 T

28 28 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Now subharmonicity of a 2 gives cδ a(λ) a(λ) 2 + b(λ) 2 2 a(λ) 2 ( a(λ) + b(λ) )2 a(λ) 2, which proves that (a, b) is a corona pair. Let us now prove that a 2 (A 2 ). From Theorem 7.6 we know that P + : L 2 ( a 2 dm) L 2 (hdm) is bounded (note that 0 a 2 δh and thus a 2 L ). On the other hand, since h a 2, the space L 2 (hdm) embeds continuously into L 2 ( a 2 dm). As a consequence, P + is bounded from L 2 ( a 2 dm) to itself which implies that a 2 (and equivalently a 2 ) satisfies the (A 2 ) condition. As a consequence of our discussion, we can deduce the following result: Theorem 8.3. Let b H be non-extreme, and µ M + (D ). Then the following are equivalent: () We have H (b) µ = H (b) and f µ f b for any f H (b); (2) The following conditions hold: (a) the function a is µ-admissible, (b) the pair (a, b) is a corona pair, (c) the function a 2 satisfies (A 2 ), (d) the measure ν, defined by dν = a 2 dµ, satisfies 0 < inf I ν (S(I)) m(i) sup I ν (S(I)) m(i) <, where the infimum and supremum above are taken over all open arcs I of T. Example 8.4. Surely an example is needed here. Let a(z) := c α ( z) α, where α (0, /2) and c α is chosen so that a H. As we have already mentioned earlier, since 0 < α < /2, the function a 2 satisfies the (A 2 ) condition. Choose b to be the outer function in H satisfying a 2 + b 2 = on T. Standard theory, using the fact that a is Hölder continuous on D, will show that b is continuous on D (see [9]). It follows that (a, b) is a corona pair. If σ M + (D ) is any Carleson measure for H 2, then one can show that dµ := a 2 dm + dσ satisfies the conditions of the above theorem.

29 DIRECT AND REVERSE CARLESON MEASURES 29 We end this section with a proof that if b is non-extreme and nonconstant then there are no isometric measures for H (b). This requires a preliminary technical result already known [34]. We provide a different proof which is slightly shorter but needs the additional assumption that b/a H 2. Recall from Section 3 that the polynomials belong to H (b) when b is non-extreme. Lemma 8.5. Let {c k } k 0 be the Taylor coefficients of the analytic function b/a and assume that b/a H 2. Then n z n 2 b = + c j 2. Proof. By (3.2) and (3.4) we need to calculate h n 2, where h n = T b/ā z n. We have 2 h n 2 2 = P b + ā zn = P z n+ b 2 a = zn+ P z n+ b 2 n a = c j 2, 2 where for the last identity, we used the fact that z n+ P z n+ is the orthogonal projection onto (z n+ H 2 ), that is the orthogonal projection onto the set of polynomials of degree at most n. Theorem 8.6. When b is non-constant and non-extreme, there are no positive isometric measures for H (b). Proof. Let us assume to the contrary that b is non-constant (and nonextreme) and that there exists a µ M + (D ) such that f b = f µ, 2 j=0 f H (b). Let us apply this identity to f = z n. First observe that z n 2 µ = z 2n dµ(z) = µ(t) + z 2n dµ(z). D D Lemma 8.5 yields n (8.7) µ(t) + z 2n dµ(z) = z n 2 b = + c j 2, n 0. D Now let n to get µ(t) = + + j=0 c j 2. j=0 2 j=0

30 30 BLANDIGNÈRES, FRICAIN, GAUNARD, HARTMANN, AND ROSS Combine this identity with the one in (8.7) to obtain z 2n dµ(z) = 0 and c j 2 = 0 D j=n+ for all n 0. In particular, the last identity for n = 0 yields b/a must be constant, or equivalently b = ka, with k C. Hence, since = a 2 + b 2 = a 2 ( + k 2 ) a.e. on T, a 2 is constant on T. But since a is outer, this forces a, and hence b, to be constant, yielding the desired contradiction. Remark 8.8. When b is constant then H (b) = H 2 with the norms differing by the multiplicative constant b 2. In this case the only isometric measure for H 2 is Lebesgue measure m. Surely this is wellknown but we include the following simple proof for the convenience of the reader. Indeed for each n N {0} = z n 2 2 = z 2n dµ + µ(t). D Clearly, letting n, we get µ(t) =. In particular, for n = 0, this yields µ(d) = 0 and µ = µ T. By Carleson s criterion we see that µ m and so dµ = hdm. To conclude that h is equal to one almost everywhere, apply the fact that µ is an isometric measure to the normalized reproducing kernels κ λ. Remark 8.9. In [5] we introduced the concept of dominating sets for H (b) when b is inner. One might be tempted to define a similar notion of dominating set for H (b) as a Borel set E T satisfying (8.0) f 2 b f 2 dm, f H (b). E Since we always have E f 2 dm f 2 b, f H (b), this implies that the norms L 2 (χ E dm) and b are equivalent on H (b). Note that in [5] it is shown that dominating sets exist for every model space (bh 2 ) and it has been used to give sufficient conditions for reverse Carleson embeddings for these spaces. When b is not inner, however, there is no real point to this because it can be proved that there are no dominating sets for H (b). The proof of this fact is based on the following technical lemma. Lemma 8.. Let b H such that f b f 2 for all f H (b). Then either b is an inner function or b <.