Bergman projections of vector-valued functions · Bergman projections of vector-valued functions...

Bergman projections of vector-valuedfunctions

Alexandru Alemanjoint with Olivia Constantin

Alexandru Aleman joint with Olivia Constantin Bergman projections of vector-valued functions

Let:D be the unit disc in the complex plane,A be the normalized area measure on D.

The Bergman projection is the orthogonal projection from L2(D)onto the closed subspace consisting of analytic functions in D.An elementary calculation shows that this singular operator isexplicitly defined for every integrable function on the disc by

P0f (z) =

∫D

1(1− ζ̄z)2

f (ζ)dA(ζ), z ∈ D. (1)


Let:D be the unit disc in the complex plane,A be the normalized area measure on D.

The Bergman projection is the orthogonal projection from L2(D)onto the closed subspace consisting of analytic functions in D.An elementary calculation shows that this singular operator isexplicitly defined for every integrable function on the disc by

P0f (z) =

∫D

1(1− ζ̄z)2

f (ζ)dA(ζ), z ∈ D. (1)


This is the Bergman-space analogue of the well known Rieszprojection on spaces of integrable functions on the unit circle T

P−1f (z) =

∫T

11− ζ̄z

f (ζ)|dζ|, z ∈ D. (2)


Even if they are connected by the chain of standard weightedBergman projections

Pαf (z) =

∫D

1(1− ζ̄z)α+2

f (ζ)dAα(ζ), z ∈ D, (3)

where α > −1 and dAα(ζ) = (α + 1)(1− |ζ|2)αdA(ζ),

P0, Pα are essentially different from the Riesz projection P−1.

While most of the properties of P−1 are due to the cancellationin the Szegö kernel, the kernels arising in Pα induce very littlecancellation. For example,

P+α f (z) =

∫D

1|1− ζ̄z|α+2

f (ζ)dAα(ζ), z ∈ D

has the same mapping properties as Pα, α > −1.


Even if they are connected by the chain of standard weightedBergman projections

Pαf (z) =

∫D

1(1− ζ̄z)α+2

f (ζ)dAα(ζ), z ∈ D, (3)

where α > −1 and dAα(ζ) = (α + 1)(1− |ζ|2)αdA(ζ),

P0, Pα are essentially different from the Riesz projection P−1.

While most of the properties of P−1 are due to the cancellationin the Szegö kernel, the kernels arising in Pα induce very littlecancellation. For example,

P+α f (z) =

∫D

1|1− ζ̄z|α+2

f (ζ)dAα(ζ), z ∈ D

has the same mapping properties as Pα, α > −1.


It is a challenge to understand the deformation of Bergmanprojections into the Riesz projection.

Maybe the family of weights should be refined. There is veryrecent work in this direction by J. A. Pelaez and J. Rättyä.


The vector-valued case

The differences between these projections become morestriking in the vector-valued case. that is if we fix a separableHilbert space H and consider Pα, P−1 acting on integrableH-valued functions f .

Let L2,αa (H), α > −1, equal the space of analytic H-valued

functions f in D with

‖f‖ =

(∫D‖f (z)‖2H dAα(z)

) 12

<∞. (4)

Similarly, when α = −1, the corresponding space ofanalytic functions is the Hardy space H2(H) with

‖f‖ = limr→1

(∫T‖f (rz)‖2H |dz|

) 12

<∞. (5)



The differences between these projections become morestriking in the vector-valued case. that is if we fix a separableHilbert space H and consider Pα, P−1 acting on integrableH-valued functions f .

Let L2,αa (H), α > −1, equal the space of analytic H-valued

functions f in D with

‖f‖ =

(∫D‖f (z)‖2H dAα(z)

) 12

<∞. (4)

Similarly, when α = −1, the corresponding space ofanalytic functions is the Hardy space H2(H) with

‖f‖ = limr→1

(∫T‖f (rz)‖2H |dz|

) 12

<∞. (5)



Then for α > −1

Pα(L2(D,H)) = L2,αa (H) ,

andP−1(L2(T,H)) = H2(H) .


Hankel operators

We want to use Hankel operators that map the Hardy andBergman spaces into themselves, therefore we introduce themvia the following bilinear forms.

Given analytic functions T : D→ B(H), and x , y : (1 + ε)D→ Hlet

〈ΓT x , y〉 = limr→1

∫T〈T (rz) x(r z̄), y(rz)〉 |dz|, (6)

and for α > −1,


∫D〈T (rz) x(r z̄), y(rz)〉dAα(z), (7)


Hankel operators

We want to use Hankel operators that map the Hardy andBergman spaces into themselves, therefore we introduce themvia the following bilinear forms.

Given analytic functions T : D→ B(H), and x , y : (1 + ε)D→ Hlet


∫T〈T (rz) x(r z̄), y(rz)〉 |dz|, (6)

and for α > −1,


∫D〈T (rz) x(r z̄), y(rz)〉dAα(z), (7)


The Hardy space case

By a deep factorization result of Sarason thecharacterization of bounded Hankel operators amounts tofinding an intrinsic characterization of Riesz projections ofbounded measurable operator-valued functions on the unitcircle.

The scalar case is resolved by the celebrated Feffermanduality theorem and the result says that this classcoincides with the set of functions f ∈ H2 whose boundaryvalues have bounded mean oscillation:

supI

1|I|

∫I|f (z)− fI ||dz| <∞, where fI =

1|I|

∫If (z)|dz|,

(8)and the supremum is taken over all arcs I on the unit circle.



By a deep factorization result of Sarason thecharacterization of bounded Hankel operators amounts tofinding an intrinsic characterization of Riesz projections ofbounded measurable operator-valued functions on the unitcircle.

The scalar case is resolved by the celebrated Feffermanduality theorem and the result says that this classcoincides with the set of functions f ∈ H2 whose boundaryvalues have bounded mean oscillation:

supI

1|I|

∫I|f (z)− fI ||dz| <∞, where fI =

1|I|

∫If (z)|dz|,

(8)and the supremum is taken over all arcs I on the unit circle.



When trying to extend this to the operator-valued settingthe difficulty is self-evident; one has find the right quantityto replace the modulus in the formula defining boundedmean oscillation.

If we choose the operator norm instead, the resultingcondition is too strong, it will not be satisfied by all Rieszprojections of bounded operator-valued functions.

The second natural choice is to consider the so-called ”so”condition

‖f‖BMOso = sup‖x‖H=1

supI

1|I|

∫I‖f (z)x − fIx‖H|dz| <∞. (9)







supI

1|I|

∫I‖f (z)x − fIx‖H|dz| <∞. (9)







supI

1|I|

∫I‖f (z)x − fIx‖H|dz| <∞. (9)



Nazarov, Pisier, Treil, Volberg (2002) proved that there ared × d matrix-symbols T with ‖T‖BMOso = ‖T ∗‖BMOso = 1 forwhich

‖ΓT‖ & log d .

This difficult open problem has a solution in an importantspecial case, the Hardy space on the polydisc Dn.Ferguson and Lacey (n = 2, 2004) and Lacey andTerwilleger (general n, 2009) proved the weak factorizationof H1(Dn) as the projective tensor product spaceH2(Dn)⊗ H2(Dn).Equivalently, it holds that a Hankel operator is bounded onH2(Dn) if and only its holomorphic symbol belongs to thedual of H1(Dn), which is the product BMO space, asidentified by Chang and Fefferman.




‖ΓT‖ & log d .





‖ΓT‖ & log d .



Infinitely many variables?

A variant of the above result for the coresponding spaces onthe infinite polydisc D∞ would be of great importance for thestudy of Dirichlet series ∑

n

ann−s .

Howeverthe constants in the work of Lacey-Terwilleger:

BMO-norm/Hankel normgo to infinity with n.

Very recently, J. Ortega and K. Seip proved that theseconstants grow exponentially in n.




n

ann−s .







n

ann−s .





The Bergman space case

The characterization of bounded Hankel operators on Bergmanspaces is the same as the well-known scalar result.

Theorem

ΓT extends to a bounded linear operator on L2,αa (H) if and only

ifsupz∈D

(1− |z|2)‖T ′(z)‖ <∞.

The proof can be obtained by imitating the scalar methods.Alternatively, one can use the duality theory of Bergmanspaces as developed by Arregui and Blasco.




Theorem


ifsupz∈D

(1− |z|2)‖T ′(z)‖ <∞.





Theorem


ifsupz∈D

(1− |z|2)‖T ′(z)‖ <∞.





Theorem


ifsupz∈D

(1− |z|2)‖T ′(z)‖ <∞.



Similarity to a contraction

An operator T on a Hilbert space H is called similar to acontraction if there exist a contraction S (‖S‖ ≤ 1) and aninvertible operator V on H such that T = V−1SV .

By von Neumann’s inequality such an operator will satisfy forany analytic polynomial p

‖p(T )‖ ≤ ‖V‖‖V−1‖ supz∈D|p(z)|,

i.e. T is polynomially bounded.

The similarity problem, i.e. whether a polynomially boundedoperator is similar to a contraction, was a long-standing openproblem in operator theory posed by Halmos in his "Tenproblems in Hilbert space" in 1970.





‖p(T )‖ ≤ ‖V‖‖V−1‖ supz∈D|p(z)|,







‖p(T )‖ ≤ ‖V‖‖V−1‖ supz∈D|p(z)|,





Attempting to find a counterexample, Peller considered thefollowing type of operators (sometimes calledFoias-Williams/Peller type operators or Foguel-Hankeloperators)

Rf =

(S∗ Γf0 S

)(10)

acting on the direct sum H2⊕H2, where S is the shift(multiplication by the independent variable) operator on H2, andΓf is the Hankel operator with symbol f .

Rf is polynomially bounded if f ′ ∈ BMO (Peller)Rf is similar to a contraction if f ′ ∈ BMO (Bourgain)If Rf is polynomially bounded then f ′ ∈ BMO (Aleksandrovand Peller)




Rf =

(S∗ Γf0 S

)(10)






Rf =

(S∗ Γf0 S

)(10)






Rf =

(S∗ Γf0 S

)(10)




In 1996 Pisier solved the similarity problem in the negative. Hisexample is an operator of the form

RT =

(S∗ ΓT0 S

)(11)

with T analytic and operator valued, acting on the direct sum ofHardy spaces with values in an infinite dimensional Hilbertspace.

The infinite dimensional version of this operator was crucial forPisier’s construction as Paulsen and Davidson showed that nosuch examples are possible in the finite dimensional case.


In 1996 Pisier solved the similarity problem in the negative. Hisexample is an operator of the form

RT =

(S∗ ΓT0 S

)(11)

with T analytic and operator valued, acting on the direct sum ofHardy spaces with values in an infinite dimensional Hilbertspace.

The infinite dimensional version of this operator was crucial forPisier’s construction as Paulsen and Davidson showed that nosuch examples are possible in the finite dimensional case.



We can of course, consider the same type of operator on theweighted Bergman spaces L2,α

a (H). The following result wasobtained by us in 2004. Fergusson and Petrovic proved thescalar version 2 years earlier.

Theorem

Let T : D→ B(H) be a holomorphic operator-valued functionwith supz∈D(1− |z|2)‖T ′(z)‖ <∞. Then the following areequivalent

(i) RT is power bounded (supn ‖RnT‖ <∞);

(ii) RT is polynomially bounded;(iii) RT is similar to a contraction;(iv) sup

z∈D(1− |z|2)‖T ′′(z)‖ <∞.



We can of course, consider the same type of operator on theweighted Bergman spaces L2,α

a (H). The following result wasobtained by us in 2004. Fergusson and Petrovic proved thescalar version 2 years earlier.

Theorem

Let T : D→ B(H) be a holomorphic operator-valued functionwith supz∈D(1− |z|2)‖T ′(z)‖ <∞. Then the following areequivalent

(i) RT is power bounded (supn ‖RnT‖ <∞);

(ii) RT is polynomially bounded;(iii) RT is similar to a contraction;(iv) sup

z∈D(1− |z|2)‖T ′′(z)‖ <∞.


The angle between past and future

The Hunt-Muckenhoupt-Wheeden theorem characterizesthe weights w for which the Riesz projection is bounded onthe weighted space L2(T,w) in terms of the A2-condition

supI

(1|I|

∫Iw)(

1|I|

∫Iw−1

)<∞.

The boundedness of the Riesz projection translates to thefact that the angle between the "past" and the "future" of astationary process with spectral measure W is nonzero.

The most obvious impediment is that the proof in the scalarcase relies heavily on the use of maximal functions, a toolthat is not available for matrix(operator)-valued weights.




supI

(1|I|

∫Iw)(

1|I|

∫Iw−1

)<∞.






supI

(1|I|

∫Iw)(

1|I|

∫Iw−1

)<∞.




Treil and Volberg (’97) overcome this difficulty by means ofwavelet techniques and obtain an analogue of thiscelebrated theorem for matrix-valued weights. The Rieszprojection is bounded on the weighted space L2(W ),where W is a d × d matrix-valued weight, if and only if

supI

∥∥∥∥∥(

1|I|

∫IW)1/2( 1

|I|

∫IW−1

)1/2∥∥∥∥∥ <∞.

This result does not generalize to the case d =∞. Againthe constants involved grow like log d (Gillespie, Pott, Treil,Volberg (2004)).


The Bergman projection in weighted spaces

The Bergman space analogue of the scalarHunt-Muckenhoupt-Wheeden theorem, was obtained byBékollé and Bonami:

Theorem

Pα is bounded on L2(D,wdAα) if and only if

supS

1A2α(S)

∫SωdAα

∫S

1ω

dAα <∞,

where the supremum is taken over all Carleson squares

S = {z = reit : 1− h < r < 1, |t − θ| < h},

with h ∈ (0,1), θ ∈ [0,2π).

The class of such weights is denoted by B2(α).


The Bergman projection in weighted spaces

The Bergman space analogue of the scalarHunt-Muckenhoupt-Wheeden theorem, was obtained byBékollé and Bonami:

Theorem

Pα is bounded on L2(D,wdAα) if and only if

supS

1A2α(S)

∫SωdAα

∫S

1ω

dAα <∞,

where the supremum is taken over all Carleson squares

S = {z = reit : 1− h < r < 1, |t − θ| < h},

with h ∈ (0,1), θ ∈ [0,2π).

The class of such weights is denoted by B2(α).


Operator-valued weights

We consider operator-valued weights W : D→ B(H) suchthat W (z) is a nonnegative operator that is invertible a.e.z ∈ D.

Our only assumptions are:1 The operator-valued integrals

∫D W±1dAα exist (i.e. they

define bounded linear operators)2∫D WdAα is invertible.




















The corresponding L2-space on D is denoted by L2(WdAα) andhas the norm

‖f‖22,W ,α =

∫D〈W (z)f (z), f (z)〉dAα(z).

It is easy to see that the subspace of L2(WdAα) consisting ofH−valued analytic functions in D is closed. We shall denotethis subspace by L2

a(WdAα).


The main result

Theorem

Let α > −1 and assume W is as above. Then the Bergmanprojection Pα is bounded on L2(WdAα) if and only if W belongsto the class B2(α), i.e. if

supS

∥∥∥∥∥(

1Aα(S)

∫S

WdAα

)1/2( 1Aα(S)

∫S

W−1dAα

)1/2∥∥∥∥∥ <∞

where the supremum is taken over all Carleson squares S in D.

Observe that the theorem holds for infinite dimensional Hilbertspace H as well!


About the proof of the sufficiency part

Assume that α = 0.Assume without loss that W is almost constant onpseudo-hyperbolic discs (i.e. replace W by its averages onsuch discs).

Prove with ”bare hands” that the Békollé-Bonami conditiondefining B2(0) implies that

P+β W (z) =

∫D

1|1− ζ̄z|β+2

W (ζ)dA(ζ) ≤ C(1−|z|2)−βW (z),

provided that β is sufficiently large (β > 4 will do in thiscase). This is a Bergman-space analogue of theA∞-condition.





P+β W (z) =

∫D

1|1− ζ̄z|β+2

W (ζ)dA(ζ) ≤ C(1−|z|2)−βW (z),






P+β W (z) =

∫D

1|1− ζ̄z|β+2

W (ζ)dA(ζ) ≤ C(1−|z|2)−βW (z),




The last estimate can be used that Pβ is bounded on L2a(W )

when β is sufficiently large.(larger than before)

But we want P0 to be bounded!

To close the gap we argue by duality. The boundedness of aprojection is equivalent to a certain representation ofcontinuous linear functionals on the space in question. Thistranslates to:















We know that when β is sufficiently large, every continuouslinear functional l on L2

a(W ) can be written as

l(f ) =

∫D〈f (z),g(z)〉H(1− |z|2)βdA(z)

with g ∈ L2a((1− |z|2)2βW−1) fixed (and unique).

We want to prove that every continuous linear functional lon L2


l(f ) =

∫D〈f (z),h(z)〉HdA(z)

with h ∈ L2a(W−1) fixed (and unique).



We know that when β is sufficiently large, every continuouslinear functional l on L2


l(f ) =

∫D〈f (z),g(z)〉H(1− |z|2)βdA(z)

with g ∈ L2a((1− |z|2)2βW−1) fixed (and unique).

We want to prove that every continuous linear functional lon L2


l(f ) =

∫D〈f (z),h(z)〉HdA(z)

with h ∈ L2a(W−1) fixed (and unique).



By Parseval’s formula (or Stokes’s) we can easily see thatif β = N is an integer then the function h we are looking foris essentially an N-th primitive of the original g.

But is it in L2a(W−1)?

(In fact we need the two norms to be comparable, but letus focus only on this question)



By Parseval’s formula (or Stokes’s) we can easily see thatif β = N is an integer then the function h we are looking foris essentially an N-th primitive of the original g.

But is it in L2a(W−1)?

(In fact we need the two norms to be comparable, but letus focus only on this question)



This will follow once we prove that for every W ∈ B2(0) wehave

‖f‖2L2a(W )

. ‖f (0)‖2 +

∫D〈Wf ′(z), f ′(z)〉(1− |z|2)2dA(z),

for all f ∈ L2a(W ).

This inequality is true! Reason: For any γ > 0

f (z) = Pγ(1− |z|2)f ′(z) + harmless terms

and we know already that Pγ is bounded on L2a(W ) when γ

is large



This will follow once we prove that for every W ∈ B2(0) wehave

‖f‖2L2a(W )

. ‖f (0)‖2 +

∫D〈Wf ′(z), f ′(z)〉(1− |z|2)2dA(z),

for all f ∈ L2a(W ).

This inequality is true! Reason: For any γ > 0

f (z) = Pγ(1− |z|2)f ′(z) + harmless terms

and we know already that Pγ is bounded on L2a(W ) when γ

is large


A word about the necessity part

This is much easier to see in the scalar case.

For λ ∈ D and α > −1 let

kλ(z) =1

(1− λ̄z)α+2, φλ(z) =

λ− z1− λ̄z

.

The key step is the elementary identity:

kz(λ) = kz(ζ)kζ(λ)

kζ(ζ)+ kz(λ)

∑n

cnφζ(λ)nφζ(z)n,

where∑

n |cn| <∞.



This is much easier to see in the scalar case.

For λ ∈ D and α > −1 let

kλ(z) =1

(1− λ̄z)α+2, φλ(z) =

λ− z1− λ̄z

.

The key step is the elementary identity:

kz(λ) = kz(ζ)kζ(λ)

kζ(ζ)+ kz(λ)

∑n

cnφζ(λ)nφζ(z)n,

where∑

n |cn| <∞.



In other words,

Pαf (λ) =∑

n

cnMφζPαM∗φζ f (λ) + Pf (ζ)kζ(λ)

kζ(ζ)

i.e. if Pα is bounded, then the rank-one term on the right isbounded.



In other words,

Pαf (λ) =∑

n

cnMφζPαM∗φζ f (λ) + Pf (ζ)kζ(λ)

kζ(ζ)

i.e. if Pα is bounded, then the rank-one term on the right isbounded.



Let f (z) = kζ(z)W−1(z) Then

‖f‖2 =

∫D|kζ(z)|2W−1(z)dAα(z)

and for this particular f the boundedness of the rank-one termon the previous slide gives the inequality

∫|kζ(z)|2W (z)dAα(z)

∫D|kζ(z)|2W−1(z)dAα(z) . kζ(ζ)2

which implies the necessity.



Let f (z) = kζ(z)W−1(z) Then

‖f‖2 =

∫D|kζ(z)|2W−1(z)dAα(z)

and for this particular f the boundedness of the rank-one termon the previous slide gives the inequality

∫|kζ(z)|2W (z)dAα(z)

∫D|kζ(z)|2W−1(z)dAα(z) . kζ(ζ)2

which implies the necessity.


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