BILINEAR FORMS ON THE DIRICHLET SPACE

Transcription

1 BILINEAR FORMS ON HE IRICHLE SPACE N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Introduction and Summary Hankel operators on the Hardy space of the disk, H 2, can be studied as linear operators from H 2 to its dual space, as conjugate linear operators from H 2 to itself, or, in the viewpoint we will take here, as bilinear functionals on H 2 H 2 In that formulation, given a holomorphic symbol function b we consider the bilinear Hankel form, defined initially for f, g in P, the space of polynomials, by S b f, g : fg, b H 2 he norm of S b is S b H 2 H 2 sup { S b f, g : f H 2 g H 2 } Nehari s classical criterion for the boundedness of S b can be cast in modern language using Fefferman s duality theorem We say a positive measure µ on the disk is a Carleson measure for H 2 if µ CMH 2 : sup { } f 2 dµ : f H 2 < and that b is in the space BMO if b BMO : b0 + b z 2 z 2 < CMH 2 Nehari s theorem [5] is the equivalence S b H 2 H b 2 BMO Our main result is an analogous statement for a similar class of bilinear forms on the irichlet space We will give the definitions and statement now with further background discussion in the next section Recall that is the Hilbert space of holomorphic functions on the disk with inner product f, g f0g0 + f zg z, and normed by f 2 f, f We consider a holomorphic symbol function b and define the associated bilinear form, initially for f, g P, by he norm of b is b f, g : fg, b b sup { b f, g : f g } NA s work partially supported by the COFIN project Analisi Armonica, funded by the Italian Minister for Research RR s work supported by the National Science Foundation under Grant No ES s work supported by the National Science and Engineering Council of Canada BW s work supported by the National Science Foundation under Grant No

2 2 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK We say a positive measure µ on the disk is a Carleson measure for if { } µ CM : sup f 2 dµ : f <, and that the function b is in the space X if b X : b0 + b z 2 < CM Our main result is heorem b b X In the next section we give some background and show how heorem can be reformulated as a duality result for a space which is presented by its weak factorization In notation introduced there we show that X We also combine our result with earlier work to conclude that I he third section contains the proof of the main theorem It is easy to see that b C b X o obtain the other inequality we must use the boundedness of b to show b 2 is a Carleson measure Analysis of the capacity theoretic characterization of Carleson measures due to Stegenga allows us to focus attention on a certain set V in and the relative sizes of V b 2 and the capacity of the set V o compare these quantities we construct V exp, an expanded version of the set V which satisfies two conflicting conditions First, V exp is not much larger than V, either when measured by V exp b 2 or by the capacity of the V exp Second, \V exp is well separated from V in a way that allows the interaction of quantities supported on the two sets to be controlled Once this is done we can construct a function Φ V which is approximately one on V and which has Φ V approximately supported on \V exp Using Φ V we build functions f and g with the property that b f, g b 2 + error V he technical estimates on Φ V allow us to show that the error term is small and the boundedness of b then gives the required control of V b 2 2 Background on Hankel Forms 2 Bilinear Hankel Forms Similar bilinear forms can be defined for many spaces of holomorphic functions Suppose H is a Hilbert space of holomorphic functions on a domain Ω and suppose for convenience that P Ω H Given a function b holomorphic in Ω and f, g P Ω we can define a Hankel form, K b, by K b f, g fg, b H One can then ask what conditions on b are necessary and sufficient for K b to be bounded hese questions have been studied in very many contexts; examples include [5], [9], [2], [6] [], [3], [20] Also, in a context not involving function theory, the boundedness result of Maz ya and Verbitsky, [4], also appears to be part of this pattern It is fascinating that although there is a great deal of variety in the techniques used in the proofs, there is a surprising similarity in the answers obtained he answer, quite generally, is that for some differential operator, b 2 can be used to define a Carleson measure for H And, although this paper

3 BILINEAR FORMS ON HE IRICHLE SPACE 3 provides another instance of this, the authors do not have a heuristic explanation for the pattern here is a close connection between boundedness results for this type of bilinear form, duality theorems for function spaces, and weak factorization results for function spaces; see for instance [] and [8] Here is how those ideas play out in this context efine the weakly factored space to be the completion of finite sums h f j g j under the norm { h inf fj g j : h } f j g j A corollary of heorem is Corollary With the pairing h, b h, b b h, we have that X hat is, if Λ, there is a unique b X with Λh b h, for h P, and Λ b b X Proof If b X and h, say h f i g i with f i g i h + ε, then h, b f i g i, b b f i, g i i i b f i g i b h + ε i It follows that Λ b h h, b defines a continuous linear functional on with Λ b b Conversely, if Λ is a continuous linear functional on with norm Λ, then Λh Λ h Λ h Λ h, for h, and so there is a unique b such that Λh Λ b h for h Finally, if h fg with f, g we have b f, g fg, b Λ b h Λh Λ h Λ f g, which shows that b extends to a continuous bilinear form on with b Λ By the theorem we conclude b X and also, with the other estimates, that Λ b b X he corresponding statement in the Hardy space is Fefferman s duality theorem, H 2 H 2 BMO However our result is only a partial analog In the Hardy space context one also has internal characterizations of the spaces involved hat is, not only is H 2 H 2 H but H can be defined intrinsically using integral means; similarly BM O can be defined using oscillation and without reference to Carleson measuresin contrast we do not have a satisfactory intrinsic characterization of either the functions in of those in X 22 Other Hankel Forms On the Hardy space, letting P denote the projection from L 2 to H 2, we have, for any f, g H 2, S b f, g fg, b H 2 f, ḡb H 2 f, P ḡb H 2 Hence the study of S b is roughly equivalent to the study of the conjugate linear map S b : g P ḡb he use of conjugate linear operators is an inessential convenience

4 4 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Similar comments apply for Hankel forms on, for example, the Bergman space However this type of rewriting does not work in because it is not true that for three holomorphic functions f, g, and b we have fg, b f, ḡb Focusing for the moment on functions which vanish at the origin, we have b f, g fg b f g + fg b f gb + fg b Let P B be the orthogonal projection of L 2, onto the Bergman space We can analyze the first summand as f gb gf b gp B f b g, I 2 P B f b Here I is the operator of indefinite integration his operator is similar in spirt to Sb but some symmetry has been lost; it is not true that f, b g equals g, b f and in fact both terms are needed to reconstruct b f, g Operators such as S b and b are also sometimes called Hankel operators It is shown in [7] that b is bounded exactly if b X < An analogous result for real variable operators is in [0] efine the space I to be the completion of the space of functions h such that h can be written as a finite sum, h f j g j,with the norm { h I inf fj g j : h } f jg j efine X 0 to be the norm closure in X of the polynomials Corollary 2 X 0 I Proof It is shown by Wu in [8] that the first two spaces are the same It is immediate that I It is shown in [7] that I X Combining that with the previous corollary we see that the second and third spaces have the same dual Hence by the Hahn-Banach theorem the two spaces agree 23 Preliminary Steps of the Proof

5 BILINEAR FORMS ON HE IRICHLE SPACE 5 23 he Easy irection Suppose that µ b is a -Carleson measure For f, g P we have b f, g f 0 g 0 b 0 + f z g z + f z g z b z f 0 g 0 b 0 + f z g z b z + f z g z b z fgb0 + f g 2 dµ b C b 0 + µ b Carleson f g C b X f g 2 + g f 2 dµ b hus b has a bounded extension to with b C b X We also note for later use that if b extends to a bounded bilinear form on then b Setting g we obtain f, b b f, b f for all polynomials f P, which shows that b and 2 b C b 232 Capacity and ree Extremal Functions For an interval I in the circle we let I m be its midpoint and zi I 2π z be the associated index point in the disk In the other direction we set Iz to be the interval such that ziz z We set I, the tent over I to be the convex hull of I and zi and let z z I : I More generally, for any open subset H of the circle, we set H I H I o complete the proof we will show that µ b given by µ b b 2 is a -Carleson measure by verifying a condition due to Stegenga [9] He works with a capacity defined by, for G in the circle, } 22 Cap G inf { ψ 2 : ψ 0 0, Re ψ z for z G and shows that µ is a Carleson measure exactly if for any finite collection of disjoint arcs {I j } N j in the circle we have 23 µ N j I j C Cap N ji j In our proof we use functions which are approximate extremals for measuring capacity, that is functions for which the equality in 22 is approximately attained We will also need to understand how the capacity of a set changes if we expand it in certain ways More precisely for I an open arc and 0 < ρ, let I ρ be the arc concentric with I having length I ρ For G open in let G ρ I G I ρ 2

6 6 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK In both our study of approximate extremals and of capacity of expanded sets we find it convenient to transfer our arguments to and from rotations of the Bergman tree associated with and to work with tree capacities instead A detailed description and properties of these trees are presented in [4] and we follow that treatment Consider a dyadic tree together with the following notation If x is an element of the tree, x denotes its immediate predecessor in If z is an element of the sequence Z, P z denotes its predecessor in Z: P z Z is the maximum element of Z [o, z we assume o Z for convenience Let θ be the rotation of the tree by the angle θ, and let Cap θ be the tree capacity associated with θ: Cap θ G inf f κ 2 : f o 0, f β for β θ, I β G κ θ We say that S is a stopping region if every pair of distinct points in S are incomparable in Let Ω A point x is in the interior of Ω if x, x, x +, x Ω A function H is harmonic in Ω if 24 Hx 3 [Hx + Hx + + Hx ] for every point x which is interior in Ω Let Ih x y [o,x] h y If H Ih is harmonic in Ω, then we have that 25 hx hx + + hx whenever x is in the interior of Ω he following proposition and remark are essentially proved in [4] Proposition Let be a dyadic tree and suppose that E and F are subsets as above here is an extremal function H Ih such that CapE, F h 2 l 2 2 he function H is harmonic on \ E F 3 If S is a stopping region in, then κ S h κ 2CapE, F 4 he function h is positive on geodesics joining a point of E to a point of F, and zero everywhere else Remark here is the following useful formula for computing capacities: given z < ζ and U ± S ζ ± S z +, 26 Capz, U + U Capζ, U + + Capζ, U + dz, ζ[capζ, U + + Capζ, U ] 233 he Capacity of ree Blowups We also need the tree analogue G ρ θ of the blowup G ρ of an open set G: G ρ θ { A ρ κ : κ θ with I κ G}, where A ρ κ denotes the unique element in the tree θ satisfying o A ρ κ κ, ρd κ < d A ρ κ ρd κ +

7 BILINEAR FORMS ON HE IRICHLE SPACE 7 Consider a subtree {o, κ, β, γ} of distinct elements in the dyadic tree where o < κ β γ and β, γ are incomparable Given ε > 0, suppose that β < β and γ < γ where d β d β + ε and d γ d γ + ε We claim that 27 Cap o {β, γ} Cap o {β, γ } + ε Indeed, with a d κ, b d κ, β, c d κ, γ and b, c defined by we have and a + b + ε a + b and a + c + ε a + c, Cap o {β, γ } Cap o {β, γ} hus we need to prove that d κ + d κ + a + b + c [b + ε a + b] [c + ε a + c] [b + ε a + b] + [c + ε a + c] bc b + c A calculation reveals that the left side is dκ,β + dκ,γ dκ,β + dκ,γ a + bc b + c, a + b c b + a + b c + c [b + ε a + b] [c + ε a + c] a + [b + ε a + b] + [c + ε a + c] Cap o {β, γ} [b + ε a + b] [c + ε a + c] + [b + ε a + b] + [c + ε a + c] bc b + c ε Cap o {β, γ} ε a + bc b + c b + c [b + ε a + b] [c + ε a + c] bc {b + c + ε 2a + b + c} {b + c + ε 2a + b + c} b + c b + c ab + ac + 2bc bc 2a + b + c ε {b + c + ε 2a + b + c} b + c ε a + b a + c +ε {b + c + ε 2a + b + c}, so that after dividing by ε and multiplying by b + c, we are left with showing that b + c ab + ac + 2bc bc 2a + b + c + ε a + b a + c b + c {b + c + ε 2a + b + c} ab + ac + bc

8 8 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK he left hand side is of the form fε α + βε / γ + δε with all the quantities positive Hence f ε is continuous and of constant sign and it suffices to verify the inequality at ε 0 and in the limit as ε At ε 0 we have b + c ab + ac + 2bc bc 2 A + b + c b + c ab + ac + bc 2abc b + c ab + ac + bc For the limit as ε we need to check a + b a + c b + c ab + ac + bc 2a + b + c or a + b a + c b + c 2a + b + c ab + ac + bc When we do the multiplication the same monomials appear on both sides and the coefficients on the left are never larger he analogue of 27 holds by the same proof for the virtual edges created in the algorithm for computing capacities in [4] Applying induction we obtain the following result Proposition 2 Let Z {z j } N j be a stopping time in Choose elements zj z j such that d zj + ε d zj for j N hen with Z { zj we have Cap o Z Cap o Z + ε } N j We will use the following corollary Given a point w and 0 < ρ <, define w ρ to be the unique point in satisfying o w ρ w and ρd w < d w ρ ρd w + Corollary 3 Let W {w l } M l be a stopping time in Suppose 0 < ρ < and let Z {z j } N j consist of the minimal tree elements in the set {wρ l }M l hen Cap o Z < ρ Cap ow Proof For each j, there is w l such that z j w ρ l Fix such a choice and let z j w l hen d z j d wl < ρ d wρ l ρ d z j for j N he corollary now follows from Proposition 2 with ρ + ε he conclusion of the corollary can be stated Cap {o}, Z < Cap {o}, W ρ Unfortunately we cannot extend these arguments in the opposite direction to obtain a condenser estimate, Cap Z, W < Cap {o}, W ρ Indeed, the left side can even be finite while the right side is finite as shown by an example below hus the geometric blowup construction does not lead to a useful capacity estimate for the condenser Cap W ρ, W On the other hand we do

9 BILINEAR FORMS ON HE IRICHLE SPACE 9 have a useful capacity estimate for W ρ in terms of CapW while achieving a good geometric separation between W ρ and W, a fact that plays a crucial role in using Schur s test to estimate an integral below, as well as in estimating an error term near the end of the paper o construct an appropriate condenser, we will instead use a method based on a capacitary extemal and a comparison principle Let W be a stopping time in By Proposition, there is a unique extremal function H Ih such that 28 H o 0, H x for x W, CapW h 2 l 2, where Cap denotes tree capacity in Given stopping times E, F we say that E F if for every x E there is y F with y < x For stopping times E F denote by G E, F the union of all those geodesics connecting a point of x E to the point y E lying above it, ie y < x Given a stopping time W, the corresponding extremal H satisfying 28, and 0 < ρ <, define the capacitary blowup W ρ of W as opposed to the geometric blowup W ρ of W by W ρ {t G {o}, W : H t ρ and H x ρ for x < t} Lemma Cap W ρ ρ 2 CapW Proof: Let H ρ ρ H and hρ ρh where h H and H is the extremal for W in 28 hen H ρ is a candidate for the infimum in the definition of capacity of W ρ, and hence by the comparison principle, 2 Cap W ρ h ρ 2 l h 2 2 l ρ 2 ρ 2 CapW his capacitary blowup W ρ, unlike the geometric blowup W ρ, does indeed satisfy a condenser inequality Note that by 222 below, it suffices to obtain a condenser inequality only for W with small capacity Lemma 2 Cap W, W ρ 4 ρ 2 CapW provided CapW 4 ρ2 Proof: Let H be the extremal for W in 28 For t W ρ we have by our assumption, and so h t h l 2 CapW ρ, 2 H t H At + h t ρ + + ρ ρ 2 2 If we define H { } t 2 ρ H t +ρ 2, then H 0 on W ρ and H on W hus H is a candidate for the capacity of the condenser and so by the comparison principle, Cap W, W ρ H 2 l 2 G W gρ,w H 2 l h 2 l ρ 2 4 ρ 2 CapW

10 0 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Corollary 4 Let Z be a stopping time in and suppose that 0 < γ < α < Set E Z α and F Ẽρ where ρ γ α hen CapF γ CapZ, Cap E, F 4 α γ 2 CapZ α Proof: We have from Lemma and Corollary 3, CapF CapẼρ ρ 2 CapE α γ CapZα < α γ and then from Lemma 2 and Corollary 3, α CapZ γ CapZ, 4 Cap E, F Cap E, Ẽρ ρ 2 CapE < 4 α ρ 2 CapZ Recall that G is the tent above G as defined in the tree, and similarly for G t, 0 < t < Given a stopping time Z we let S Z be the shadow of Z on the circle, so that 0 S Z Z Consider the condenser E, F where E 0 G α {wk α} k and F Ẽ γ α {w γ k } k Note that each point wα k in E is a descendent of a unique w γ l in F By Proposition, there is a unique extremal function H Ih such that H 0 on F, H on E, Cap E, F h 2 l 2, where Cap denotes tree capacity in It follows from Corollary 4 that 29 Cap G Cap E, F 4 α γ 2 Cap G α Notation Let 0 < β < γ < α < We will use the shorthand notations V α G G α, γ α V γ G Ṽ G α, V β β G S γ V γ G We also define maximal intervals {J α k } k, {J γ k } k, {J β k } k such that V α G J α k and V γ G J γ k and V β G J β k

11 BILINEAR FORMS ON HE IRICHLE SPACE hus VG α is a geometric blowup of G, V γ G is a capacitary blowup of V G α, and V β G is again a geometric blowup of V γ G We have from above the estimates Cap V α G < α CapG, Cap V γ G < γ CapG, Cap V β G < β CapG, Cap V α G, V γ G < C α,γcapg Remark 2 he geometric blowups have good geometric separation properties useful when estimating Bergman type kernels while the capacitary blowup has a good condenser estimate useful in constructing holomorphic extremals he geometric separation between VG α and G is used in 225 below in conjunction with the Schur test, the good condenser estimate of VG α, V γ G is used in the construction of a holomorhic extermal in Lemma 3 below, and finally the geometric separation between V β G and V γ G is used in estimating term 4 ABA near the end of the paper 234 Holomorphic Approximate Extremals and Capacity Estimates Now we define a holomorphic approximation Φ to the function H Ih on constructed in Proposition We will use a parameter s In addition to specific assumptions made at various places we will always assume efine ϕ κ z κ 2 κz +s and s > 20 Φ z κ h κ ϕ κ z κ h κ Note that and so κ h κ Iδ κ z I κ h κ δ κ κ 2 +s κz z Ih z H z, 2 Φ z H z κ h κ {ϕ κ Iδ κ } z We will also need to write Φ in terms of the projection operator ζ 2 s 22 Γ s h z h ζ +s, ζz namely as Φ Γ s g where 23 g ζ κ h κ +s ζκ B κ ζ 2 s χ Bκ ζ, and B κ is the Euclidean ball centered at κ with radius c κ for a sufficiently small positive constant c to be chosen later he function Φ satisfies the following estimates

12 2 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Lemma 3 Suppose z and s > hen we have Φ z Φ wk α CCap E, F, z J k α Re Φ w 24 k α c > 0, k M α Φ wk α C, k M α Φ z CCap E, F, z / V γ G Lemma 4 Furthermore, if s > 2 then Φ Γ sg where 25 g ζ 2 CCap E, F Corollary 5 For s > 2, 25 and Lemma 24 of [5] show that 26 Φ 2 g ζ 2 CCap E, F Remark 3 In the case that VG α J α consists of a single arc, we may divide the function Φ z by Φ wk α to obtain a holomorphic function that is close to on VG α and small outside V γ G as in Lemma 4 in [5] Proof From 2 we have Φ z H z κ [o,z] I z + II z h κ {ϕ κ z } + κ / [o,z] h κ ϕ κ z We also have that h is nonnegative and supported in V γ G \ V G α We first show that II z κ 2 +s h κ CCap E, F κz For A > let κ/ [o,z] { } Ω k κ : A k κ 2 < κz A k If we choose A sufficiently close to, then for every k the set Ω k is a stopping time for, ie any two distinct elements in Ω k are incomparable in Now we use the stopping time property 3 in Proposition to obtain κ Ω k h κ CCap E, F, k 0 Altogether we then have II z k0 κ Ω k h κ A k+s C s Cap E, F If z \ V γ G, then I z 0 and H z 0 and we have Φ z Φ z H z II z C s Cap E, F, which is the fourth line in 24 If z VG α, let k be such that z wk α hen for κ / [o, z] we have ϕ κ w α k C ϕ κ z,

13 BILINEAR FORMS ON HE IRICHLE SPACE 3 and for κ [o, z] we have ϕ κ z ϕ κ wk α κ 2 +s κ 2 +s κz κwk α C z wk α s κ 2 hus Φ z Φ w α k κ [o,w α k ] C s κ [o,w α k ] C s Cap E, F, h κ ϕ κ z ϕ κ w α k + C h κ z wα k 2 + CII z κ κ / [o,z] h κ ϕ κ z since h κ CCap E, F and κ [o,wk α ] his proves the first κ 2 wk α 2 line in 24 Moreover, we note that for s 0 and κ [o, wk α], Re ϕ κ w α k Re κ 2 κw α k κ 2 Re κw α κwk α k c > 0 2 A similar result holds for s > provided the Bergman tree is constructed sufficiently thin depending on s It then follows from κ [o,wk α ] h κ that Re Φ wk α h κ Re ϕ κ wk α + h κ Re ϕ κ wk α We trivially have κ [o,w α k ] c κ [o,w α k ] Φ w α k I z + II z C κ / [o,z] h κ CCap E, F c > 0 κ [o,w α k ] h κ + CCap E, F C, and this completes the proof of 24 Finally we prove 25 From property of Proposition we obtain 2 g ζ 2 +s ζκ h κ B κ κ ζ 2 s χ Bκ ζ h κ 2 ζκ 2+2s κ B κ 2 B κ ζ 2 2s κ h κ 2 Cap E, F Corollary 6 Let G be a finite union of arcs in the circle hen 27 Cap G Cap G, where Cap denotes Stegenga s capacity on the circle

14 4 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Proof he inequality follows easily from Lemma 3 which provides a candidate for testing the Stegenga capacity of G Indeed, let c, C be the constants in Lemma 3, and suppose that Cap E, F c 3 C Set Ψ z 3 c Φ z Φ 0 hen Ψ 0 0, and by 26 we have Re Ψ z 3 {Re Φ z Re Φ 0} c 3 c {c 2C Cap E, F }, z V α G, Ψ 2 3 c 2 Φ 2 3 c 2 CCap E, F Continuing by invoking Corollary?? we obtain that for G, Ψ 2 3 c 2 CCap E, F CCap G Conversely, to obtain the inequality, let ψ be an extremal function for Cap G efine h o 0 and h κ κ ψ z dλ z, κ \ {o}, Qκ where Q h κ is the hyperbolic cube corresponding to κ in, and dλ z is invariant measure on the disk One easily verifies as in [2] that Ih o 0, Ih 2 B 2 h 2 l 2 κ 2 ψ z dλ z κ Qκ C κ Qκ ψ z C ψ 2, 2 and Ih β h κ Re ψ β c > 0, for S β G, κ [o,β] since if B h κ, R is the hyperbolic ball of radius R about κ, then for R large enough, ψ β ψ κ ψ κ κ [o,β] κ [o,β] C κ [o,β] C κ [o,β] B h κ, κ 2 B h κ, B h κ, B h κ,r ψ z ψ z B h κ, B h κ, κ 2 ψ z dλ z C h κ, Qκ κ [o,β] ψ z

15 BILINEAR FORMS ON HE IRICHLE SPACE 5 where the final inequality is the submean value property for ψ z It follows that } Cap G inf { H 2B2 : H 0 0, Re H κ if S κ G 2 c Ih C c 2 ψ 2 C c 2 Cap G B 2 Lemma 24 of Bishop [7] says that 28 Cap N j I ρ j Cρ Cap N j I j, for a constant C ρ depending only on ρ < In the next Corollary we use the uniform versions of this, ie C ρ as ρ, given by Proposition 2 and its corollaries Set dσ dσ/2π on Now let G be an open set in such that 29 µ b θ G dσ Cap θ G dσ M : sup E open µ b θ E dσ Cap θ E dσ Corollary 7 With M as in 29 we have µ b 2 Carleson M Proof Using Corollary 6 and θ E E, we have M C sup µ b E dσ E open Cap E dσ C sup E open µ b E Cap E µ b 2 Carleson, where the final comparison is Stegenga s theorem Conversely, one can verify using the argument in 22 above that for 0 < ρ <, µ b E C µ b θ E ρ dσ CM Cap θ E ρ dσ CMCap E ρ CMCap E, where the third line uses 27 with E ρ and θ in place of G and, and the final inequality follows from 28 hus from Stegenga s theorem we obtain µ b 2 Carleson sup E open µ b E Cap E CM We need to know that µ b V β G \ V G is small Proposition 3 Given ε > 0 we can find β βε < to that 220 µ b V β G \ V G εµ b V G,

16 6 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Proof Corollary 3 shows that Cap θ G ρ θ ρ Cap θ G, 0 θ < 2π,and if we integrate on we obtain Cap θ G ρ θ dσ Cap θ G dσ ρ From 29 we thus have µ b θ G ρ θ dσ M Cap θ G ρ θ dσ M Cap θ G dσ ρ µ b θ G dσ ρ It follows that µ b θ G ρ θ \ θ G dσ µ b θ G ρ θ dσ µ b θ G dσ ρ µ b θ G dσ Now with σ ρ+ 2 halfway between ρ and, 22 µ b θ G ρ θ \ θ G dσ dµ b z dσ 2 θ G ρ θ\ θ G θ G ρ θ\ G { 2π G σ \ G dµ b z dσ {θ:z θ G ρ θ\ G} dµ b z, dσ } dµ b z since every z G σ lies in θ G ρ θ for at least half of the θ s in [0, 2π Here we may assume that the components of G ρ have small length since otherwise we trivially have Cap θ G dσ c > 0 and so then 222 M dµ b c c b 2 C c b 2 Combining the above inequalities and using σ ρ+ 2, 2 ρ <, we obtain µ b G σ \ G 2 ρ µ b θ G dσ 2 ρ µ b θ G dσ 8 3 σ µ b θ G dσ,

17 BILINEAR FORMS ON HE IRICHLE SPACE 7 for 3 4 σ < Recalling V G σ Gσ and V G G this becomes µ b VG σ \ V G 8 3 σ µ b θ G dσ 8 3 σ µ 3 b V G, 4 σ <, since θ G G V G for all θ hus given ε > 0 it is possible to select β to that 220 holds 235 Schur Estimates and a Bilinear Operator on rees We begin with a bilinear version of Schur s well known theorem heorem 2 Let X, µ, Y, ν and Z, ω be measure spaces and H x, y, z be a nonnegative measurable function on X Y Z efine f, g x H x, y, z f y dν y g z dω z, x X, Y Z at least initially for nonnegative functions f, g hen if < p <, is bounded from L p ν L p ω to L p µ if and only if there are positive functions h, k and m on X, Y and Z respectively such that H x, y, z k y p m z p dν y dω z Ah x p, Y Z for µ-ae x X, and X H x, y, z h x p dµ x Bk y m z p, for ν ω-ae y, z Y Z Moreover, operator AB Proof he necessity which we don t use is a standard iteration argument For the sufficiency, we have f x p dµ x X X Y Z Y Z p/p H x, y, z k y p m z p dν y dω z H x, y, z p f y g z dν y k y m z f y p dω z dµ x p p A p H x, y, z h x p g z dµ x dν y dω z Y Z X k y m z p p A p B p k y p m z p f y g z dν y dω z Y Z k y m z AB p f y p dν y g z p dω z Y Z his theorem can be used along with the estimates w 2 t C t if c < 0, t > t+c dw C t log z 2 if c 0, t >, wz C t z 2 c if c > 0, t > to prove the following Corollary which we will use later It is proved as heorem 20 in [2]

18 8 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Corollary 8 efine f z z 2 a Sf z z 2 a w 2 b f w dw, 2+a+b wz w 2 b f w dw, 2+a+b wz where the kernel of S is the modulus of the kernel of Suppose that t R and p < and set dν t z z 2 t hen is bounded on L p, dν t if and only if S is bounded on L p, dν t if and only if 224 pa < t + < p b + We now apply heorem 2 to prove a lemma about a bilinear operator mapping l 2 A l 2 B to L 2 where A and B are subsets of which are well separated Lemma 5 Suppose A and B are subsets of, h l 2 A and k l 2 B, and 2 < α < Suppose further that A and B satisfy the separation condition: κ A, γ B we have 225 κ γ γ 2 α hen the bilinear map of h, k to functions on the disk given by h, b z h κ κ 2 +s κz 2+s b γ γ 2 +s γz +s κ A is bounded from l 2 A l 2 B to L 2 Proof We will verify the hypotheses of the previous theorem he kernel function here is H z, κ, γ κ 2 +s γ 2 +s κz 2+s +s, z, κ A, γ B, γz with Lebesgue measure on, and counting measure on A and B We will take as Schur functions h z z 2 4, k κ κ 2 4 and m γ γ 2 ε 2, on, A and B respectively, where ε > 0 will be chosen sufficiently small later We must then verify κ s γ 2 +ε+s 226 κz 2+s γz +s A 2 z 2 2, for z, and 227 κ A γ B κ 2 +s κz 2+s B 2 κ 2 2 γ 2 ε, γ B γ 2 +s γz +s z 2 2

19 BILINEAR FORMS ON HE IRICHLE SPACE 9 for κ A and γ B o prove 226 we write κ s γ 2 +ε+s κz 2+s γz +s κ A γ B κ s γ 2 +ε+s κz 2+s γz +s κ A γ B hen from 223 we obtain and κ s κz 2+s C κ A γ 2 +ε+s γz +s C γ B w 2 2 +s wz 2+s dw C ζ V G which yields 226 he proof of 227 will use 225 We have κ 2 +s γ 2 +s κz 2+s z γ γ z κ κ 2 + I + II + III + IV + V By 225 κ γ ζ 2 +ε+s z 2 2 ζz +s C, γz +s z 2 2 γ 2 z γ 2 κ γ γ 2 α and so κ 2 z κ 2 κ γ + z γ, z κ κ γ I κ 2 +s κ γ 2+s κ 2 +s γ z γ γ 2 κ γ 2+s C z 2 2 κ 2 2 γ α

20 20 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Similarly we have κ 2 +s γ 2 +s II κ γ 2+s κ 2 +s γ 2 +s γ 2 z γ 2 κ γ κ γ 2+s γ 2 2 s κ 2 +s γ Continuing to use κ γ III and similarly, for some ε > 0 Finally V κ γ 2+s C κ 2 2 γ 2 +s γ 2 α we obtain κ γ +s C z γ, z κ κ γ IV C κ 2 2 γ 2 ε, κ 2 +s z κ 2+s κ 2 +s γ 2 +s z 2 2 z γ +s κ 2 2 γ α κ 2 2 γ 2 +s α, γ 2 +s z γ +s z 2 2 κ γ s C κ 2 2 γ 2 +s α 24 he Main Bilinear Estimate o complete the proof we will show that µ b is a -Carleson measure by verifying Stegenga s condition 23; that is, we will show that for any finite collection of disjoint arcs {I j } N j in the circle we have µ b N j I j C Cap N ji j In fact we will see that it suffices to verify this for the single set G N ji j described in 29 We will prove the inequality 228 µ b V G C b 2 Cap G Once we have this Corollary 6 yields M µ b θ G dσ Cap θ G dσ µ b V G Cap θ G dσ C b 2 By Corollary 7 µ b 2 Carleson M which then completes the proof of heorem

21 BILINEAR FORMS ON HE IRICHLE SPACE 2 We now turn to the proof of the estimate 228 Let 2 < β < β < γ < α < to be chosen later We will obtain our estimate by using the boundedness of b on certain functions f and g in he function f will be, approximately, b χ VG and the function g will be constructed using an approximate extremal function of the type described in Subsection 234 and will be approximately equal to χ VG We have chosen G ; we now set where E {w α k } Mα k and F {wγ l }Mγ l G α Mα k J α k and G γ Mγ l J γ l Now construct a Bergman tree for the disk with uniform bounds independent of G such that E F and both E and F are stopping times in Note that every point w α k in E is contained in a unique successor set S wγ l for a point wγ l in F Now define Φ as in 20 above, so that we have the estimates in heorem 3 and Corollary 5 From Corollaries 6 and?? we obtain 229 Cap E, F CCap G We will use g Φ 2 and 230 f z Γ s + s ζ χ V G b ζ z as our test functions in the bilinear inequality 23 b f, g b f g From 230 we have f z V G b ζ ζ 2 s +s ζz + s ζ, so that f z V G b z b ζ ζ 2 s 2+s ζz \V G b z + Λb z, b ζ ζ 2 s 2+s ζz where 232 Λb z \V G b s ζ ζ 2+s, ζz by the reproducing property of the generalized Bergman kernels ζ 2 s Now ζz 2+s if we plug f and g Φ 2 as above in b f, g we obtain b f, g b f, Φ 2

22 22 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK b fφ, Φ which we analyze as 233 b f, Φ 2 b fφ, Φ {f z Φ z + 2f z Φ z} Φ z b z + f 0 Φ 0 2 b 0 f 0 Φ 0 2 b 0 + b z 2 Φ z 2 +2 Φ z Φ z f z b z + Λb z b zφ z rivially, we have 234 C b 2 Cap E, F C b 2 Cap E, F Now we write b z 2 Φ z 2 { } + + b z 2 Φ z 2 V G V β G \V G \V β G 2 A + 2 B + 2 C he main term 2A satisfies A µ b V G + b z 2 Φ z 2 V G µ b V G + O b 2 Cap E, F, by 24 and 2 For term 2B we use 220 to obtain B Cµ b V β G \ V G Cεµ b V G Using 24 once more, we see that term 2C satisfies C b z 2 C α,β,ρ Cap E, F 3 C b 2 Cap E, F \V β G Altogether, using 234, 235, 236, 237 and 238 in 233 we have 239 µ b V G b f, Φ 2 +Cµb V β G \ V G +C b 2 Cap E, F We estimate 3 using Cauchy-Schwarz with ε > 0 small as follows: 3 2 Φ z b z Φ z f z ε Φ z b z 2 + C Φ z f z 2 ε 3 A + 3 B

23 BILINEAR FORMS ON HE IRICHLE SPACE 23 Using the decomposition and argument surrounding term 2 we obtain { } A ε + Φ z b z 2 Cε V G + V β G \V G \V β G µ b V G + C b 2 Cap E, F o estimate term 3 B we use f z Γ s + s ζ χ V G b ζ z ζ 2 s ζz +s b ζ where γ V G b γ 2 V G γ V G γ V G γ V G γ 2 +s γz +s γ 2 +s B γ b ζ γz +s b γ, ζ 2 dλ ζ b ζ 2 ζ 2 2 dλ ζ B γ V G b ζ 2 We now use the separation of \ VG α and V G he facts that A supp h \ VG α and B V G V G insure that 225 is satisfied Hence we can use Lemma 5 and the representation of Φ in 20 to continue with 3 B Φ z f z 2 C h κ 2 b γ 2, γ B κ A We also have from 2 and Corollary 5 that h κ 2 b γ 2 CCap E, F b 2 γ B κ A Altogether we then have 24 3 B CCap E, F b 2, and thus also ε b z 2 + C b 2 Cap E, F V G We begin our estimate of term 4 by 4 Λb z b zφ z b z Φ z 2 Λb z Φ z 2

24 24 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK Now we claim the following estimate for 4 A ΦΛb L 2 : A Φ z Λb z 2 Cµ b V β G \ V G + C b 2 Cap E, F εµ b V G + C b 2 Cap E, F Indeed, the second inequality follows from 220 From 232 we obtain 4 A Corollary 8 shows that { } Φ z 2 b ζ ζ s s V β G \V G \V β G ζz C Φ z 2 b ζ ζ s 2 V β G \V G ζz 2+s +C Φ z 2 b ζ ζ s 2 2+s \V β G ζz 4 AA + 4 AB 4 AA C V β G \V G V β G \V G b ζ ζ s 2 ζz 2+s b ζ 2 Cµ b V β G \ V G We write the second integral as 4 AB { V γ G } + Φ z 2 b ζ ζ s 2 2+s \V γ G \V β G ζz 4 ABA + 4 ABB, where by Corollary 8 again, 4 ABB CCap E, F 2 C b 2 Cap E, F 2 C b 2 Cap E, F b ζ 2 Finally, with β < β < γ < α <, Corollary 8 shows that the term 4 ABA satisfies the following estimate Recall that V γ G J γ k and wγ j z J γ k We set

25 BILINEAR FORMS ON HE IRICHLE SPACE 25 { } A l k : J γ k J β l and define l k by the condition k A lk hen b ζ ζ s 2 4 ABA C V γ G \V β G ζz 2+s C J γ k J γ k b ζ ζ s 2 k ζw γ 2+s k C J γ k b ζ ζ s 2 J β k J β lk lk J γ k ζw γ 2+s k C k A l J γ k b ζ ζ s 2 l J β l J β l \V β G ζz 2+s C V β G εγ β b ζ ζ s 2 V β G \V β G ζz 2+s C V β εγ β b 2 C b 2 Cap E, F G his completes the proof of 244 Now we can estimate term 4 by 4 Λb z b zφ z b z Φ z 2 Λb z Φ z A /ε 4 A Cµ b V G + C b 2 Cap E, F εµ b V G + C b 2 Cap E, F εµ b V G + C µ b V G b 2 Cap E, F +C b 2 Cap E, F, using 244 and the estimate 240 for 3 A already proved above Finally, we estimate b f, Φ 2 b fφ, Φby b fφ, Φ b Φ Φf C b Cap E, F Φf Now Φf 2 C Φ z f z 2 + C Φ z f z 2 C 3 A + C 3 B + C Φ z Λb z 2 Cµ b V G + C b 2 Cap E, F,

26 26 N ARCOZZI, R ROCHBERG, E SAWYER, AN B WICK by 244 and the estimates 240 and 24 for 3 A and 3 B When we plug this into the previous estimate we get b f, Φ 2 C b 248 Cap E, F µ b V G + b 2 Cap E, F C b 2 Cap E, F µ b V G + b Cap E, F 2 Using Proposition 3 and the estimates 242, 246 and 248 in 239 we obtain µ b V G εµ b V G + C b 2 Cap E, F +C b 2 Cap E, F µ b V G εµ b V G + C b 2 Cap E, F Absorbing the first term on the right side, and using 229, we finally obtain which is 228 µ b V G C b 2 Cap E, F C b 2 Cap G, References [] A Aleman, title***???, opics in Complex Analysis and Operator heory, Girela, Alvarez, *** eds, [2] N Arcozzi, R Rochberg and E Sawyer, Carleson measures for analytic Besov spaces, Rev Mat Iberoamericana , [3] N Arcozzi, R Rochberg and E Sawyer, Carleson measures and interpolating sequences for Besov spaces on complex balls, Mem A M S 859, 2006 [4] N Arcozzi, R Rochberg and E Sawyer, Onto interpolating sequences for the irichlet space, manuscript, 2007 [5] B Böe, Interpolating sequences for Besov spaces, J Functional Analysis, , [6] A Bonami, M Peloso and F Symesak, On Hankel operators on Hardy and Bergman spaces and related questions Actes des Rencontres d Analyse Complexe Poitiers- Futuroscope, 999, 3 53, Atlantique, Poitiers, 2002 [7] C J Bishop, Interpolating sequences for the irichlet space and its multipliers, preprint 994 [8] W Cohn, S Ferguson, and R Rochberg, Boundedness of higher order Hankel forms, factorization in potential spaces and derivations, Proc London Math Soc , no, 0 30 [9] R Coifman, R Rochberg and G Weiss, Factorization theorems for Hardy spaces in several variables Ann of Math , no 3, [0] R Coifman and Murai, Commutators on the potential-theoretic energy spaces ohoku Math J , no 3, [] S Ferguson and M Lacey, A characterization of product BMO by commutators, Acta Math, 89, 2002, 2, [2] S Janson, J Peetre, and R Rochberg, Hankel forms and the Fock space Rev Mat Iberoamericana 3 987, no, 6 38 [3] M Lacey and E erwilleger, Hankel Operators in Several Complex Variables and Product BMO, to appear, Houston J Math [4] V Maz ya and I Verbitsky, he Schrödinger operator on the energy space: boundedness and compactness criteria Acta Math , no 2, [5] Z Nehari, On bounded bilinear forms Ann of Math , [6] V Peller, Hankel operators and their applications, Springer Monographs in Mathematics Springer-Verlag, New York, 2003 [7] R Rochberg and Z Wu, A new characterization of irichlet type spaces and applications, Ill J Math , 0-22

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