Corona Theorems for Multiplier Algebras on B n

Corona Theorems for Multiplier Algebras on B n Brett D. Wick Georgia Institute of Technology School of Mathematics Multivariate Operator Theory Banff International Research Station Banff, Canada August 16th, 010 B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 1 / 30

This is joint work with: Şerban Costea McMaster University Canada Eric T. Sawyer McMaster University Canada B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory / 30

Talk Outline Talk Outline Motivation of the Problem Review of the Corona Problem in One Variable Besov-Sobolev Spaces and Multiplier Algebras Baby Corona versus Corona Main Result and Sketch of Proof Further Results and Future Directions B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 3 / 30

Motivations for the Problem The Corona Problem for H (D) The Banach algebra H (D) is the collection of all analytic functions on the disc such that f H (D) := sup f (z) < z D To each z D we can associate a multiplicative linear functional on H (D) (point evaluation at z) ϕ z (f ) := f (z) Let denote the maximal ideal space of H (D). (Maximal ideals = kernels of multiplicative linear functionals) B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 4 / 30

Motivations for the Problem The Corona Problem for H (D) In 1941, Kakutani asked if there was a Corona in the maximal ideal space of H (D), i.e. whether or not the disc D was dense in. One then defines the Corona of the algebra to be \ D. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 5 / 30

Motivations for the Problem The Corona Problem for H (D) In 196, Lennart Carleson demonstrated the absence of a Corona by showing that if {g j } N is a finite set of functions in H (D) satisfying 0 < δ g j (z) 1, z D, then there are functions {f j } N in H (D) with f j (z) g j (z) = 1, z D. Moreover, f j H (D) C(δ). B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 6 / 30

Motivations for the Problem Extensions of the Corona Problem The point of departure for many generalizations of Carleson s Corona Theorem is the following: Observation H (D) is the (pointwise) multiplier algebra of the classical Hardy space H (D) on the unit disc. Namely, let M H (D) denote the class of functions ϕ such that ϕf H (D) C f H (D), f H (D). ( ) with ϕ MH (D) = inf{c : ( ) holds}. Then ϕ H (D) if and only if ϕ M H (D) and, ϕ MH (D) = ϕ H (D). B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 7 / 30

Motivations for the Problem Besov-Sobolev Spaces The space B σ (B n) is the collection of holomorphic functions f on the unit ball B n such that { m 1 k=0 f (k) (0) + where dλ n (z) = B n ( 1 z ) m+σ f (m) (z) dλ n (z) } 1 <, ( 1 z ) n 1 dv (z) is the invariant measure on B n and m + σ > n. Various choices of σ give important examples of classical function spaces: σ = 0: Corresponds to the Dirichlet Space; σ = 1 : Drury-Arveson Hardy Space; σ = n : Classical Hardy Space; σ > n : Bergman Spaces. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 8 / 30

Motivations for the Problem Besov-Sobolev Spaces The spaces B σ (B n) are examples of reproducing kernel Hilbert spaces. Namely, for each point λ B n there exists a function k λ B σ (B n) such that f (λ) = f, k λ B σ It isn t too difficult to compute (or show) that the kernel function k λ is given by 1 k λ (z) = ( ) σ 1 λz σ = 1 : Drury-Arveson Hardy Space; k λ(z) = 1 1 λz σ = n : Classical Hardy Space; k 1 λ(z) = (1 λz) n σ = n+1 : Bergman Space; k 1 λ(z) = (1 λz) n+1 B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 9 / 30

Motivations for the Problem Multiplier Algebras of Besov-Sobolev Spaces M B σ (B n ) We are interested in the multiplier algebras, M B σ (B n ), for B σ(b n). A function ϕ belongs to M B σ (B n ) if ϕf B σ (B n) C f B σ (B n) f B σ (B n ) ϕ MB σ (B n) = inf{c : above inequality holds}. It is easy to see that M B σ (B n ) = H (B n ) X σ(b n). Where X σ(b n) is the collection of functions ϕ such that for all f B σ(b n): f (z) ( 1 z ) m+σ ϕ (m) (z) dλ n (z) C f B σ (Bn), ( ) B n with ϕ X σ (B n) = inf{c : ( ) holds}. Thus, we have ϕ MB σ (B n) ϕ H (B n) + ϕ X σ (B n). B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 10 / 30

Motivations for the Problem The Corona Problem for M B σ (B n ) We wish to study a generalization of Carleson s Corona Theorem to higher dimensions and additional function spaces. Question (Corona Problem) Given g 1,..., g N M B σ (B n ) satisfying 0 < δ N g j (z) 1 for all z B n. Does there exist a constant C n,σ,n,δ and functions f 1,..., f N M B σ (B n ) satisfying f j MB σ (B n) C n,σ,n,δ g j (z) f j (z) = 1, z B n? B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 11 / 30

Motivations for the Problem The Baby Corona Problem It is easy to see that the Corona Problem for M B σ (B n ) implies a simpler question that one can consider. Question (Baby Corona Problem) Given g 1,..., g N M B σ (B n ) satisfying 0 < δ N g j (z) 1 for all z B n and h B σ (B n). Does there exist a constant C n,σ,n,δ and functions f 1,..., f N B σ (B n) satisfying f j B σ(bn) C n,σ,n,δ h B σ(bn), g j (z) f j (z) = h (z), z B n? B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 1 / 30

Main Results Baby Corona Theorem for B σ p (B n ) Theorem (Ş. Costea, E. Sawyer, BDW (Analysis & PDE 010)) Let 0 σ and 1 < p <. Given g 1,..., g N M B σ p (B n ) satisfying 0 < δ g j (z) 1, z B n, there is a constant C n,σ,n,p,δ such that for each h B σ p (B n ) there are f 1,..., f N B σ p (B n ) satisfying f j p Bp σ(bn) C n,σ,n,p,δ h p Bp σ(bn), g j (z) f j (z) = h (z), z B n. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 13 / 30

Main Results The Corona Theorem for M B σ (B n ) Corollary (Ş. Costea, E. Sawyer, BDW (Analysis & PDE 010)) Let 0 σ 1 and p =. Given g 1,..., g N M B σ (B n ) satisfying 0 < δ g j (z) 1, z B n, there is a constant C n,σ,n,δ and there are functions f 1,..., f N M B σ (B n ) satisfying f j MB σ (B n) C n,σ,n,δ g j (z) f j (z) = 1, z B n. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 14 / 30

Main Results The Corona Theorem for M B σ (B n ) The proof of this Corollary follows from the main Theorem very easily. When 0 σ 1 the spaces Bσ (B n) are reproducing kernel Hilbert spaces with a complete Nevanlinna-Pick kernel. By the Toeplitz Corona Theorem, we then have that the Baby Corona Problem is equivalent to the full Corona Problem. The result then follows. An additional corollary of the above result is the following: Corollary For 0 σ 1, the unit ball B n is dense in the maximal ideal space of M B σ (B n ). This is because the density of the the unit ball B n in the maximal ideal space of M B σ (B n ) is equivalent to the Corona Theorem above. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 15 / 30

Sketch of Proofs Sketch of Proof of the Baby Corona Theorem Given g 1,..., g N M B σ p 0 < δ (B n ) satisfying g j (z) 1, z B n, Set ϕ j (z) = g j (z) Pj g h(z). We have that N j (z) g j(z)ϕ j (z) = h(z). This solution is smooth and satisfies the correct estimates, but is far from analytic. In order to have an analytic solution, we will need to solve a sequence of -equations: For η a -closed (0, q) form, we want to solve the equation ψ = η for ψ a (0, q 1) form. To accomplish this, we will use the Koszul complex. This gives an algorithmic way of solving the -equations for each (0, q) with 1 q n after starting with a (0, n) form. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 16 / 30

Sketch of Proofs Sketch of Proof of the Baby Corona Theorem This produces a correction to the initial guess of ϕ j, call it ξ j, and set f j = ϕ j ξ j. By the Koszul complex we will have that each f j is in fact analytic. Algebraic properties of the Koszul complex give that j f jg j = h. However, now the estimates that we seek are in doubt. To guarantee the estimates, we have to look closer at the solution operator to the -equation on -closed (0, q) forms. Following the work of Øvrelid and Charpentier, one can compute that the solution operator is an integral operator that that takes (0, q) forms to (0, q 1) forms with integral kernel: (1 wz) n q ( 1 w ) q 1 Here (w, z) = 1 wz (w, z) n (w j z j ) 1 q n. ( 1 w ) ( 1 z ). B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 17 / 30

Sketch of Proofs Sketch of Proof of the Baby Corona Theorem One then needs to show that these solution operators map the Besov-Sobolev spaces B p σ(b n ) to themselves. This is accomplished by a couple of key facts: The Besov-Sobolev spaces are very flexible in terms of the norm that one can use. One need only take the parameter m sufficiently high. We show that these operators are very well behaved on real variable versions of the space B p σ(b n ). These, of course, contain the space that we are interested in. To show that the solution operators are bounded on L p (B n ; dv ) the original proof uses the Schur Test. To handle the boundedness on B p σ(b n ), we can also use the Schur test but this requires more work to handle the derivative. This is key to the proof. Certain properties of the unit ball (i.e., symmetries) are exploited to make some of these computations easier. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 18 / 30

Sketch of Proofs More Detailed Sketch Charpentier s Solution Operators Charpentier proves the following formula for (0, q)-forms: Theorem (Charpentier (1980)) For z B n and 0 q n 1 and smooth forms of degree (0, q + 1): { } f (z) = C q f (ξ) Cn 0,q+1 (ξ, z) + c q z f (ξ) Cn 0,q (ξ, z). B n B n Here, C 0,q n C 0,q n (ξ, z) is a (n, n q 1)-form in ξ and a (0, q)-form in z with (w, z) = ( 1) q Φ q n (w, z) sgn (ν) (w iν z iν ) dw j dz l ωn (w j J ν l L ν ν P q n ( (1 wz) n 1 q 1 w ) q Φ q n (w, z) (w, z) n, 0 q n 1. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 19 / 30

Sketch of Proofs More Detailed Sketch Charpentier s Solution Operators We can solve z u = f for a -closed (0, q + 1)-form f as follows: Set u(z) c q f (ξ) Cn 0,q (ξ, z) B n Taking z of this we see Charpentier and f = 0 that ( ) z u = c q z f (ξ) Cn 0,q (ξ, z) = f (z). B n B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 0 / 30

Sketch of Proofs More Detailed Sketch Application of the Koszul Complex We must show that f = Ω 1 0 h Λ g Γ 0 ( Bσ p Bn ; l ) where Γ 0 antisymmetric -tensor of (0, 0)-forms that solves is an Γ 0 = Ω 1h Λ g Γ 3 1. Inductively, we will have Γ q+ q (0, q)-forms that solves is an alternating (q + )-tensor of Γ q+ q = Ω q+ q+1 h Λ g Γ q+3 q+1, up to q = n 1. Since Γ n+ n = 0 and the (0, n)-form Ω n+1 n is -closed. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 1 / 30

Sketch of Proofs More Detailed Sketch Application of the Koszul Complex Using the Charpentier solution operators C 0,q n,s on (0, q + 1)-forms and the Koszul complex we then get f = Ω 1 0h Λ g Γ 0 = Ω 1 0h Λ g Cn,s 0,0 ( 1 Ω 1 h Λ g Γ 3 ) 1 The goal is to establish. = Ω 1 0h Λ g C 0,0 n,s 1 Ω 1h + Λ g C 0,0 n,s 1 Λ g C 0,1 n,s Ω 3 h + + ( 1) n Λ g C 0,0 n,s 1 Λ g C 0,n 1 F 0 + F 1 + + F n. n,s n Ω n+1 n f B σ p (B n;l ) C n,σ,p,δ h B σ p (B n). h B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory / 30

Sketch of Proofs More Detailed Sketch Schur s Lemma on Besov-Sobolev Spaces Lemma Let a, b, c, t R. Then the operator T a,b,c f (z) = B n (1 z ) a (1 w ) b ( (w, z) ) c 1 wz n+1+a+b+c f (w) dv (w) is bounded on L (B p n ; (1 w ) ) t dv (w) if and only if c > n and pa < t + 1 < p (b + 1). We will use this Lemma for appropriate choices of a, b, c. This, plus some more computations, shows that the it is possible to obtain estimates to in the space B σ p (B n ). B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 3 / 30

Sketch of Proofs More Detailed Sketch Obtaining the Estimates We will accomplish this by showing F µ B σ p (B n;l ) C n,σ,p,δ h B σ p (B n), 0 µ n. Then we can then estimate by F µ B σ p (B n;l ) T1 T T µ Ω µ+1 µ B σ p (B n;l ). Here T µ = T aµ,b µ,c µ for appropriate a µ, b µ, c µ. So, we have F µ B σ p (B n;l ) C n,σ,p,δ B σ p (B n;l ). Ω µ+1 µ Finally, show that for Ω µ+1 µ arising from the Koszul complex, then B σ p (B n;l ) C n,σ,p,δ h B σ p (B. n) Ω µ+1 µ B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 4 / 30

Further Results and Questions The H (B n ) Corona Problem When σ = n (the classical Hardy space H (B n )), we have a weaker version of the Corona Problem that we can prove. Theorem (Ş. Costea, E. Sawyer, BDW (J. of Funct. Anal. 010)) Given g 1,..., g N H (B n ) satisfying 0 < δ N g j (z) 1 for all z B n. Then there is a constant C n,n,δ and there are functions f 1,..., f N BMOA(B n ) satisfying f j BMOA(Bn) C n,n,δ g j (z) f j (z) = 1, z B n. This gives another proof of a famous theorem of Varopoulos. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 5 / 30

Further Results and Questions Baby Corona for H (B n ) versus Corona for H (B n ) We know that the Corona Problem always implies the Baby Corona Problem. By the Toeplitz Corona Theorem, we know that, under certain conditions on the reproducing kernel, these problems are in fact equivalent. But, what happens if we don t have these conditions? Theorem (Equivalence between Corona and Baby Corona, (Amar 003)) Let {g j } N H (B n ). Then there exists {f j } N H (B n ) with f j (z)g j (z) = 1 z B n and g j H (B n) 1 δ if and only if M µ g (M µ g ) δ I µ for all probability measures µ on B n. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 6 / 30

Further Results and Questions Baby Corona for H (B n ) versus Corona for H (B n ) This suggests how to attack the Corona Problem for H (B n ). Difficulty is that one must solve the Baby Corona Problem for every probability measure on B n. Instead, it is possible to reduce this to a class of probability measures for which the methods of harmonic analysis might be are more amenable. Theorem (Trent, BDW (Complex Anal. Oper. Theory, 009)) Let H {H H (B n ) : H nonvanishing in B n, 1 H L ( B n, dσ), and H = 1}. Assume that M H g M H g δ I H for all H H. Then there exists a f 1,..., f N H (B n ), so that f j (z)g j (z) = 1 z B n and f j H (B n) 1 δ. Reduces the H (B n ) Corona Problem to a weighted Baby Corona Problem. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 7 / 30

Further Results and Questions Open Problems and Future Directions 1 Does the algebra H (B n ) of bounded analytic functions on the ball have a Corona in its maximal ideal space? Does the Corona Theorem for the multiplier algebra of the Drury-Arveson space B 1 (B n ) extend to more general domains in C n? 3 Can we prove a Corona Theorem for any algebra in higher dimensions that is not the multiplier algebra of a Hilbert space with the complete Nevanlinna-Pick property? Any 1 < σ n would be extremely interesting. 4 Can one prove the equivalence between a weakened version of the Baby Corona Problem and the Corona Problem when 1 < σ < n? This would be useful to approach the above problem. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 8 / 30

Internet Analysis Seminar Announcement Internet Analysis Seminar Announcement 1st Internet Analysis Seminar will take place during the Fall 010 Spring 011 academic year. Phase I (October February), approximately fifteen weekly, electronic lectures will be provided via a public website. Phase II (March May), participants from Phase I will be organized into smaller, diverse groups from various institutions to work on more advanced projects. Phase III consists of a final one-week workshop held in July, during which, teams will present their projects and additional lectures will be delivered by leading experts in the field. Topic: Dirichlet Space of Analytic Functions. Details available on: http://people.math.gatech.edu/ bwick6/internetanalysisseminar.html Supported by National Science Foundation CAREER grant DMS # 095543. B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 9 / 30

Conclusion Thank You! B. D. Wick (Georgia Tech) Corona Theorems Multi. Operator Theory 30 / 30