Bilinear Forms on the Dirichlet Space

Transcription

1 Bilinear Forms on the irichlet Space Brett. Wick University of South Carolina epartment of Mathematics 17th St. Petersburg Meeting in Mathematical Analysis Euler International Mathematical Institute St. Petersburg, Russia June 23rd, 2008 B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 1 / 1

2 This is joint work with: Nicola Arcozzi University of Bologna Italy Richard Rochberg Washington University in St. Louis United States of America Eric T. Sawyer McMaster University Canada B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 2 / 1

3 Talk Outline Talk Outline Motivation and History of the Problem Review of irichlet Space Theory Hankel Forms on the irichlet Space Main Result and Sketch of Proof Corollaries, Further Results and Questions B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 3 / 1

4 Motivations for the Problem Bilinear Forms on the Hardy Space The Hardy space H 2 () is the collection of all analytic functions on the disc such that f 2 H := sup f (rξ) 2 dm(ξ) < 2 0<r<1 The Hankel Operator H b maps H 2 () to H 2 () and is given by T H b := (I P H 2) M b To study the boundedness of this operator, we can study only the corresponding bilinear Hankel form T b : H 2 () H 2 () C, T b (f, g) := fg, b H 2 B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 4 / 1

5 Motivations for the Problem Bilinear Forms on the Hardy Space Lemma The bilinear form T b is bounded if and only if b belongs to BMOA(). We can connect this to Carleson measures for the space H 2 (). A function b BMOA() if and only if b H 2 () and is a Carleson measure for H 2 (). Theorem b (z) 2 (1 z 2 )da(z) The bilinear form T b : H 2 () H 2 () C is bounded if and only if is a Carleson measure for H 2 (). b (z) 2 (1 z 2 )da(z) B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 5 / 1

6 Review of irichlet Space Theory The irichlet Space 2 () efinition (irichlet Space) An analytic function is an element of the irichlet space 2 () if and only if f 2 := f (0) 2 + f (z) 2 da(z) < 2 In terms of Fourier coefficients one has the equivalent norm given by f 2 n ˆf (n) 2 2 n=0 The following inclusion relations hold 2 () H 2 () A 2 () B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 6 / 1

7 Review of irichlet Space Theory Carleson Measures for 2 () efinition A measure µ on is a 2 ()-Carleson measure if and only if f (z) 2 dµ(z) C(µ) 2 f 2 2 for all f 2 (). The best constant in the above embedding is called the norm of the Carleson measure C(µ) := µ 2 Carleson. This is a function theoretic quantity. We also want a geometric quantity that we can use to study Carleson measures. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 7 / 1

8 Review of irichlet Space Theory Carleson Measures for 2 () Logarithmic Capacity on the isc For an interval I T, let T (I ) be the Carleson tent over the interval I, { T (I ) := z : 1 I z 1, z } z I This definition obviously extends to general compact sets E T. Given a compact subset E T, the capacity of the set E is defined by cap(e) := inf { f 2 2 : Re f 1 on T (E) }. It is easy to see that for an interval I T we have cap(i ) ( ( )) 2π 1 log I B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 8 / 1

9 Review of irichlet Space Theory Carleson Measures for 2 (): Geometric Characertization There is an obvious necessary condition a 2 ()-Carleson must satisfy: Suppose that µ is a 2 -Carleson measure. For λ, let k λ (z) := 1 + log 1 1 λz. Then k λ 2 () and k λ 2 log ( 1 λ 2). Let k 2 λ denote the (approximately) normalized version of k λ. For each interval I T there exists a unique λ with 1 λ 2 = I. Standard estimates show: µ (T (I )) cap(i ) k λ (z) 2 dµ(z) C(µ) 2 k λ 2 2 C(µ) 2. Analogue of the Carleson measure condition for H 2 (). Unfortunately, this simple condition is not sufficient. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 9 / 1

10 Review of irichlet Space Theory Carleson Measures for 2 (): Geometric Characertization Theorem (Stegenga (1980)) A measure µ is a Carleson measure for 2 () if and only if ( ) ) µ N j=1t (I j ) S(µ) cap ( N j=1i j, for all finite unions of disjoint arcs on the boundary T. This is a geometric characterization of the Carleson measures. But, it is difficult to check: Computing capacity is hard. One has to check every possible collection of disjoint intervals in T. One has an equivalence between the quantities C(µ) and S(µ), C(µ) 2 S(µ). B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 10 / 1

11 Review of irichlet Space Theory Hankel Operators on the irichlet Space Hankel Operators on the irichlet Space In an analogous manner, one defines the (small) Hankel operator h b : 2 () 2 () by h b := P 2M b = b (z)f (z)g(z)da(z) efinition Suppose that b is analytic on. It belongs to X () if and only if f (z) 2 b (z) 2 da(z) C 2 f 2, f 2 (). ( ) 2 Moreover, b X := inf{c : ( ) holds} + b(0) Namely, dµ b (z) := b (z) 2 da(z) is a 2 ()-Carleson measure and b X = µ b 2 Carleson + b(0). B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 11 / 1

12 Review of irichlet Space Theory Hankel Operators on the irichlet Space Hankel Operators on the irichlet Space Theorem (Rochberg, Wu (1993)) Suppose that b is analytic on. Then h b is bounded if and only if b X (). Moreover, h b 2 2 µ b 2 Carleson. One can also look at the corresponding problem for the bilinear form T b : 2 () 2 () C. But, one can easily observe that the operator h b doesn t induce the bilinear form T b. Conjecture The bilinear form T b : 2 () 2 () C is bounded if and only if b X (). B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 12 / 1

13 Results and Proofs Main Result Theorem (N. Arcozzi, R. Rochberg, E. Sawyer, BW (2008)) Let T b : 2 () 2 () C be the bilinear form defined by T b (f, g) := fg, b 2 = f (0)g(0)b(0) + b (z) ( f (z)g(z) + f (z)g (z) ) da(z). Let dµ b (z) := b (z) 2 da(z). Then T b is a bounded bilinear form on 2 () 2 () if and only if b X () with b X := µ b 2 Carleson + b(0) T b 2 2 C. This Theorem demonstrates that the corresponding picture for the Hardy space H 2 () carries over to 2 (). B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 13 / 1

14 Results and Proofs Proof of Main Result: Easy irection Carleson Measure Bounded Bilinear Form Suppose that µ b is a 2 ()-Carleson measure. For f, g Pol () we have T b (f, g) := f (0)g(0)b(0) + b (z) ( f (z)g(z) + f (z)g (z) ) da(z) T b (f, g) f (0)g(0)b(0) + f (z)g(z)b (z) da(z) + f (z)g (z)b (z) da(z) B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 14 / 1

15 Results and Proofs Proof of Main Result: Easy irection Carleson Measure Bounded Bilinear Form T b (f, g) f (0)g(0)b(0) + f 2 ( ( g(z) 2 dµ b (z) ) g 2 f (z) 2 dµ b (z) ( b(0) + µ b 2 Carleson) f 2 g 2 = b X f 2 g 2. So, T b has a bounded extension from 2 () 2 () C with T b 2 2 C b X. ) 1 2 B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 15 / 1

16 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Choose an (almost) extremal collection of intervals {I j } j T so that we have ) ) µ b ( N j=1 T (I j) µ b ( N j=1 T (I j) S(µ b ) := sup cap( N j=1 I = j) cap( N j=1 I j) This method of proof was suggested to us by Michael Lacey. We will use this collection of intervals to construct functions f and g to test in the bilinear for T b. We will prove an estimate of the form: ) µ b ( N j=1 T (I j) cap( N j=1 I T b 2 j) 2 2 C. The function g will be constructed using an approximate extremal function from the collection of intervals that achieves the supremum and will be approximately equal to the indicator function on N j=1 T (I j). The function f will be, approximately, b on the set N j=1 T (I j). B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 16 / 1

17 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Trees on There is a discrete version of the irichlet space that can be used simple model. To construct the dyadic tree T, first form the Whitney decomposition. The center of each box is a vertex on the tree. The origin of is the root of the tree, o. We say that a vertex α is a child of β if the arc on T corresponding to α is a child of the arc corresponding to β. One then defines the dyadic irichlet Space as B 2 (T ) := {f : T C : f (o) 2 + α T f (α) 2 := f 2 B 2 < } One can recover results on 2 () from results on B 2 (T ) by averaging, mean value properties, etc. The theory of the dyadic irichlet spaces and the connection with 2 () has been deeply explored by Arcozzi, Rochberg and Sawyer. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 17 / 1

18 bg=white Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 18 / 1

19 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Using the dyadic tree T, and the extremal intervals we selected, we can form a holomorphic function ϕ that is basically the indicator of j T (I j ). Lemma There exists a holomorphic function ϕ such that ϕ(z) ϕ(wk α ) cap( N j=1 I j), z T (Ik α) Re ϕ(wk α) c > 0, 1 k M α ϕ(wk α ) C, 1 k M α( ϕ (z) cap( N j=1 I j), z / N j=1 T I γ j ). Moreover, ) ϕ 2 cap ( N j=1i 2 j. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 19 / 1

20 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure We will use g = ϕ 2 and f (z) := N j=1 T (I j ) b (ζ) ( ) da(ζ) 1 ζz ζ Using the reproducing kernel property we find that f b (ζ) (z) = ( ) 2 da(ζ) 1 ζz N j=1 T (I j ) = b (z) \ N j=1 T (I j ) =: b (z) + Λb (z) b (ζ) ( 1 ζz ) 2 da(ζ) This function f is approximately b on the set N j=1 T (I j) B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 20 / 1

21 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure If we substitute these into the bilinear form T b we find that: T b (f, g) = T b (f, ϕ 2 ) = T b (f ϕ, ϕ) { = f (z)ϕ(z) + 2f (z)ϕ (z) } ϕ(z)b (z)da(z) +f (0)ϕ(0) 2 b(0) = f (0)ϕ(0) 2 b(0) + b (z) 2 ϕ(z) 2 da(z) +2 ϕ(z)ϕ (z)f (z)b (z)da(z) + Λb (z)b (z)ϕ(z) 2 da(z) := (1) + (2) + (3) + (4). B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 21 / 1

22 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Term (1) is trivial. Term (2) yields (using properties of ϕ and a key geometric property) that (2) = = { b (z) 2 ϕ(z) 2 da(z) + N j=1 T (I j ) =: (2 A ) + (2 B ) + (2 C ). The main term (2 A ) satisfies ) (2 A ) = µ b ( N j=1t (I j ) N j=1 T (I β j )\ N j=1 T (I j ) + ) = µ b ( N j=1t (I j ) + O N j=1 T (I j ) } + b (z) 2 ϕ(z) 2 da \ N j=1 T (I β j ) ( T b 2 cap b (z) 2 ( ϕ(z) 2 1 ) da(z) ( )) N j=1t (I j ). Terms (2 B ) and (2 C ) are error terms and are controlled by properties of ϕ. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 22 / 1

23 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Terms (3) and (4) are error terms. Using properties of ϕ, geometric estimates, and Schur s Lemma, we can show these are controlled by estimates of the form ( ɛµ b n j=1 T (I j ) ) ( ) + C(ɛ) T b C cap N j=1i j where ɛ > 0 is a small number to be chosen later. Thus, we have µ b ( n j=1 T (I j ) ) ɛµ b ( n j=1 T (I j ) ) + C(ɛ) T b C cap ( N j=1i j ) Choosing ɛ > 0 small enough gives the result. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 23 / 1

24 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Key Observations If T b extends to a bounded bilinear form on 2 () 2 () C then b 2 (). Setting g = 1 we obtain: f, b 2 = T b (f, 1) T b 2 2 C f 2 for all polynomials f Pol().This implies that b 2 () and b 2 T b 2 2 C. Proposition Given ε > 0 we can find β = β(ε) < 1 so that ( ) ( ) µ b N(β) j=1 T (I β j ) \ N j=1t (I j ) εµ b N j=1t (I j ) B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 24 / 1

25 Results and Proofs Proof of Main Result: Hard irection Bounded Bilinear Form = Carleson Measure Proof. Note that ) ( µ b ( N j=1t (I j ) + µ b N(β) j=1 T (I β ) j ) \ N j=1t (I j ) ( ) = µ b N(β) j=1 T (I β j ) ( ) S(µ b ) cap N(β) j=1 I β j. Since {I j } N j=1 is the maximal collection of intervals in the geometric definition of 2 ()-Carleson measures. Next observe that ( ) ) cap N(β) j=1 I j (1 + ε) cap ( N j=1i j. This immediately gives that ( ) µ b N(β) j=1 T (I β j ) \ N j=1t (I j ) ) ε cap ( N j=1i j. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 25 / 1

26 Further Results and Questions Corollaries of The Main Result There is a close connection between boundedness of the bilinear form, duality theorems for function spaces, and weak factorization results. efinition The weakly factored space 2 () ˆ 2 () is the completion of finite sums h = f j g j under the norm { h 2 ˆ 2 = inf fj 2 g j 2 : h = } f j g j. Corollary With the pairing (h, b) = h, b 2 = T b (h, 1) we have that ( 2 () ˆ 2 ()) = X (). Namely, for Λ ( 2 () ˆ 2 () ), there is a unique b X with Λh = T b (h, 1) for h Pol (), and Λ = T b 2 2 C b X. B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 26 / 1

27 Further Results and Questions Further Results and Questions Further Results: Without much difficulty one can characterize the symbols of bounded bilinear forms that are bounded on Bα() p B q α () C. One should be able to use these ideas to prove the corresponding inequality on the unit ball in C n. (etails still need to be checked!) Questions: Can one give an intrinsic characterization of the space X ()? Equivalent question: Can one give an intrinsic characterization of the space 2 () ˆ 2 ()? Can one prove the corresponding bilinear inequality for the spaces p () q ()? Can one prove the corresponding bilinear inequality for the spaces 2 α() 2 α() C? B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 27 / 1

28 Conclusion Thank You! B.. Wick (University of South Carolina) Bilinear Forms on the irichlet Space St. Petersburg Meeting 28 / 1