Weighted Bergman projections on the Hartogs triangle: exponential decay

New York Journal of Mathematics

New York J. Math. 22(2016) 1271–1282.

Weighted Bergman projections on the Hartogs triangle: exponential decay

Liwei Chen and Yunus E. Zeytuncu

Abstract. We study weighted Bergman projections on the Hartogs triangle inC². We show that projections corresponding to exponentially vanishing weights have degenerateL^pmapping properties.

Contents

1. Introduction 1271

2. Proof of Theorem 1.1 1275

3. Proof of Theorem 1.2 1279

4. Proof of Theorem 1.3 1281

References 1281

1. Introduction

Let Ω⊂Cⁿ be a bounded domain andµbe a nonnegative function on Ω.

We sayµis an admissible weight if L²_a(Ω, µ), the space of square integrable holomorphic functions, is a closed subspace ofL²(Ω, µ), the space of square integrable functions with respect to µ(z)dV(z) where dV(z) stands for the Lebesgue measure. The weighted Bergman projectionB^µ_Ω is the orthogonal projection operator from L²(Ω, µ) ontoL²_a(Ω, µ). It is an integral operator of the form

B^µ_Ωf(z) = Z

Ω

B_Ω^µ(z, w)f(w)µ(w)dV(w)

forf ∈L²(Ω, µ). We refer to [Kra01] and [PW90] for basic definitions and properties. The analytic properties of the operatorB^µ_Ω and kernelB^µ_Ω(z, w) depend on the geometry of the domain Ω and the function theoretic prop- erties of the weightµ.

One particular investigation is relating theL^p mapping properties of B^µ_Ω and the order of vanishing of the weight µ on the boundary of Ω. One

Received February 17, 2016.

2010Mathematics Subject Classification. Primary 32A25; secondary 32A07, 32A36.

Key words and phrases. Weighted Bergman projection, exponential weights, Hartogs triangle.

The work of the second author was partially supported by a grant from the Simons Foundation (#353525).

ISSN 1076-9803/2016

1271

LIWEI CHEN AND YUNUS E. ZEYTUNCU

way of defining weights that vanish on the boundary is to take different compositions of a distance to the boundary function. In particular, let δ be a distance to the boundary function and ν(x) : [0,∞) → [0,∞) be a continuous function that only vanishes atx= 0. Then the compositionν(δ) is an admissible weight on Ω and one can study the L^p regularity of B^ν(δ)_Ω and relate it to the order of vanishing ofν(x) at 0. Below we go over some known results using this notation.

We start with the unit disc D inCand choose δ(z) = 1− |z|². Then we list the following results.

• Ifν ≡1 orν(x) =x^k for some k >0, then the weighted projection operatorB^ν(δ)

D is bounded fromL^p(D, ν(δ)) to itself for allp∈(1,∞).

See [Zhu91,FR75].

• On the other hand, ifν(x) = exp −¹_x

then the weighted projection operatorB^ν(δ)

D is bounded from L^p(D, ν(δ)) to itself only for p= 2.

See [Dos04,Zey13].

This change inL^p mapping properties between the polynomial vanishing and the exponential vanishing has been detected on some Reinhardt domains too. In particular, let Ω be a smooth bounded complete Reinhardt domain of finite type inC² and let ρ be a smooth multi-radial defining function for Ω. We choose δ=−ρ. Then we recall the following results.

• If ν ≡ 1 or ν(x) = x^t for some rational number t > 0, then the weighted projection operator B^ν(δ)_Ω is bounded from L^p(Ω, ν(δ)) to itself for allp∈(1,∞). See [McN94,CDM14,CDM15].

• On the other hand, ifν(x) = exp −¹_x

then the weighted projection operatorB^ν(δ)_Ω is bounded from L^p(Ω, ν(δ)) to itself only forp= 2.

See [CZ16b].ˇ

In addition to these contrasting results, there are more results in the polynomial decay case where theL^p boundedness on the full interval (1,∞) is observed. See for example [BG95] and [CL97]. Therefore, the degenerate L^p regularity for exponential weights on other domains arises as a natural question. More specifically, we pose the following question.

Question. Let Ω be a bounded domain with some additional geometric properties (e.g., finite type, convex, etc.) andδ be a distance to the bound- ary function. For ν(x) = exp −_x¹

, is the weighted projection operator B^ν(δ)_Ω bounded onL^p(Ω, ν(δ)) for anyp6= 2?

In this note, we investigate this question on the Hartogs triangle. In C², the Hartogs triangle His given by

H=

(z1, z2)∈C² : |z₂|<|z₁|<1 .

The source of many counterexamples onHis the singular point at the origin.

Hence, the natural choice of a distance function here is the distance to the

singular point, that is, we set

δ(z) =|z₁|.

For this choice ofδ, the polynomial decay case is already studied in [Che13, Che17, CZ16a]. Although the L^p boundedness does not hold for all p ∈ (1,∞), there is always a nondegenerate interval around p = 2 where the weighted projection operator isL^p bounded.

Theorem ([Che17]). Let δ(z) =|z₁| and ν(x) = x^t where t ∈(0,∞) with the unique decompositiont=s+ 2ksuch thatk∈Zands∈(0,2]. Then the weighted Bergman projection B^ν(δ)

H is bounded on L^p(H, ν(δ))if and only if p∈

t+4

s+k+1,_k+1^t+4 .

Note that for any t >0, there is an interval around 2 where the weighted Bergman projection is L^p bounded. Moreover, this interval gets smaller as t gets bigger. In other words, higher order vanishing of ν(x) indicates smaller L^p boundedness range. In the light of this observation we answer the question above as follows.

Theorem 1.1. Let ν(x) = exp −¹_x

, δ(z) =|z₁|, and define the exponen- tially decaying weight

µ(z) =ν(δ(z)) = exp

− 1

|z₁|

. Then the weighted Bergman projection B^µ

H is bounded on L^p(H, µ) if and only ifp= 2.

Although we state the result for a single choice of ν(x), it will be clear in the proof that the statement holds for more general choices of the form ν(x) = x^sexp −¹_x

for s ∈ R. Moreover, further generalizations can be formulated on domains that are variants of the Hartogs triangle, see [Che13]

and [EM16a].

In addition to the change in L^p regularity of the weighted Bergman pro- jection, another related change takes place in the boundary behavior of the weighted Bergman kernel with exponentially decaying weights. For poly- nomially decaying weights on the Hartogs triangle, the weighted Bergman kernel on the diagonal does not grow faster than a power of the distance function. The following result can be derived from [Che17, Lemma 3.1], where the author presents a detailed study of polynomial weights on the Hartogs triangle.

Theorem ([Che17]). Let δ(z) =|z₁| and ν(x) = x^t where t ∈(0,∞) with the unique decomposition t= s+ 2k such that k ∈Z and s ∈(0,2]. Then there exist real numbers τ and C such that

δ(z)^τB^ν(δ)

H (z, z)≤C

LIWEI CHEN AND YUNUS E. ZEYTUNCU

as z approaches the origin inside any cone

V_γ ={(z₁, z₂)∈C² : γ|z₂|<|z₁|}, where γ >1.

However, the weighted Bergman kernel with respect to exponential weights grows faster than any power of the distance to the boundary function. We state the following result.

Theorem 1.2. Let ν(x) = exp −¹_x

, δ(z) =|z₁|, and define the exponen- tially decaying weight

µ(z) =ν(δ(z)) = exp

− 1

|z₁|

. Then for any τ >0,

δ(z)^τB^ν(δ)

H (z, z)

is unbounded as z approaches the origin inside any cone V_γ ={(z₁, z₂)∈C² : γ|z₂|<|z₁|}, where γ >1.

Furthermore, if we insert a factor of µ(z) into the product then we get boundedness. See the remark after the proof of Theorem1.2.

Finally, we demonstrate another change from polynomial decay to expo- nential decay in the setting of generalized Hartogs triangles. Recently, in a series of papers [EM16a,Edh16],L^pboundedness of the Bergman projection operator is studied on the following variants of the Hartogs triangle. Let k be a positive number and define

Hk={(z₁, z₂)∈C² : |z₂|<|z₁|^k<1}.¹

It is shown that when k is a positive integer, the (unweighted) Bergman projection is bounded on L^p(Hk) for some values ofp.

Theorem ([EM16a,EM16b]). For positive integerk, the Bergman projec- tion B_Hk is bounded on L^p(Hk) if and only if p∈

2k+2 k+2,^2k+2_k

.

On the other hand, the domain Hk will become pieces of boundaries as k tends to ∞. In order to study the degenerate L^p boundedness, we define the following exponential version as the limiting domain inC²

H∞=

(z₁, z₂)∈C² : |z₂|<exp

− 1

|z₁|

, 0<|z₁|<1

. Then we prove the following result.

Theorem 1.3. The Bergman projectionB_H∞ is bounded onL^p(H∞) if and only ifp= 2.

1In [EM16a,Edh16], authors denote this domain byH¹_k.

Recently, Edholm and McNeal [EM16b] obtained similar degenerate L^p regularity on domainsHγ where γ is an irrational number. These examples are particularly interesting since the degeneracy is not due to the exponential decay but to nonrationality of the singularity.

In the following sections, we present proofs for the three statements above.

We start with Theorem 1.1, where the proof is based on some asymptotic computations. Then we continue with Theorem1.2 and we present a proof by obtaining an (almost) explicit closed form for the weighted Bergman kernel on the diagonal. Finally, the proof of Theorem 1.3 is based on the weighted theory on the punctured disc.

We highlight again that the irregular behavior in Theorems 1.1 and 1.2 are due to the exponential decay of the weight and similar results may hold on other domains. We plan to study these analogous questions on more general domains in future.

Throughout the paper, we write x≈ y to mean that there exists C > 0 such that _C¹x≤y≤Cx.

2. Proof of Theorem 1.1

We prove Theorem1.1by studying weighted Bergman projections on the punctured unit disc D^∗. Let Bλ be the weighted Bergman projection from L²(D^∗, λ) onto L²_a(D^∗, λ), where λ(z) = exp

−_|z|¹

, z ∈ D^∗. Given any sufficiently large j ∈Z⁺, we pick p > 2 +²_j. We study the behavior of the sequence

z^−jkz^k _k∈

Z⁺ under the projection B_λ. First, forα ∈R, we define

I(α) = Z 1

0

r^αe^−1rdr.

Since λis radial on D^∗ and exponential decaying at the origin, all {zⁿ}_n∈Z are orthogonal and in L²_a(D^∗, λ). So the weighted Bergman kernel has the form

B_λ(z, ζ) =

∞

X

n=−∞

c_n(zζ)ⁿ, wherec_n=

R

D^∗|z|²ⁿexp −_|z|¹

dA(z)−1

. Then by using the orthogonality of monomials and the labelling above, we get

kB_λ(z^−jkz^k)k^p_λ kz^−jkz^kk^p_λ (1)

= R

D^∗

R

D^∗B_λ(z, ζ)·ζ^−jkζ^k·exp

− _|ζ|¹ dA(ζ)

p

exp

−_|z|¹ dA(z) R

D^∗|z^−jkz^k|^pexp

−_|z|¹ dA(z)

LIWEI CHEN AND YUNUS E. ZEYTUNCU

= R

D^∗

R

D^∗

P∞

n=−∞c_n(zζ)ⁿ·ζ^−jkζ^k·exp

−_|ζ|¹ dA(ζ)

p

exp

−_|z|¹ dA(z) R

D^∗|z|^−(j−1)pkexp

−_|z|¹ dA(z)

= R

D^∗|ζ|^−2jk ·exp

−_|ζ|¹ dA(ζ)

pR

D^∗|z|^−(j+1)pkexp

− _|z|¹ dA(z)

R

D^∗|z|^−2(j+1)kexp

− _|z|¹ dA(z)

pR

D^∗|z|^−(j−1)pkexp

−_|z|¹ dA(z)

= [I(−2jk+ 1)]^p·I(−(j+ 1)pk+ 1) [I(−2(j+ 1)k+ 1)]^p·I(−(j−1)pk+ 1).

Our goal is to show that the fraction in the last line blows up. We accomplish this by studying the asymptotic behavior of the integralI(α).

Lemma 2.1. We have the following estimates on I(α) as α→ ±∞.

α→+∞lim (α+ 1)I(α) =e⁻¹, (2)

α→+∞lim

I(−α) Γ(α−1) = 1.

(3)

Proof. To show (2), we first apply the Monotone Convergent Theorem and conclude

α→+∞lim I(α) = 0.

By integration by parts, we see that I(α) =

Z 1 0

r^αe^−1r dr

= 1

α+ 1r^α+1e^−1r

1

0

− Z 1

0

1

α+ 1r^α−1e^−1r dr

= e⁻¹

α+ 1−I(α−1) α+ 1 .

By clearing the denominators, and letting α→+∞, we arrive at (2).

To show (3), we make change of variables in the definition of I(α) by x= ¹_r. We see that

I(−α) = Z ∞

1

x^α−2e^−xdx

= Γ(α−1)− Z 1

0

x^α−2e^−xdx,

where Γ is the Gamma function. Again by the Monotone Convergent The- orem

α→+∞lim Z 1

0

x^α−2e^−xdx= 0, and the fact that

α→+∞lim Γ(α−1) = +∞,

we obtain (3).

Now we combinte the asymptotic estimate in (3) and the Stirling’s formula Γ(x+ 1)≈√

2πx x

e x

asx→ ∞, to get

k→+∞lim

kB_λ(z^−jkz^k)k^p_λ kz^−jkz^kk^p_λ

= lim

k→+∞

[Γ(2jk−2)]^p·Γ((j+ 1)pk−2) [Γ(2(j+ 1)k−2)]^p·Γ((j−1)pk−2)

= lim

k→+∞

s j

j+ 1 p

j+ 1 j−1

· (2jk−3)^(2jk−3)p[(j+ 1)pk−3][(j+1)pk−3]

[2(j+ 1)k−3][2(j+1)k−3]p[(j−1)pk−3][(j−1)pk−3]

≈ lim

k→+∞

2jk−3 2(j+ 1)k−3

(2jk−3)p

·

(j+ 1)pk−3 (j−1)pk−3

(j−1)pk−3

·

(j+ 1)pk−3 2(j+ 1)k−3

2pk

.

A straightforward limit computation indicates

k→+∞lim

2jk−3 2(j+ 1)k−3

(2jk−3)p

·

j+ 1 j

(2jk−3)p

= lim

k→+∞

1− 3

2j(j+ 1)k−3j

(2jk−3)p

= exp

− 3p j+ 1

.

Similarly, we also get

k→+∞lim

(j+ 1)pk−3 (j−1)pk−3

(j−1)pk−3

·

j−1 j+ 1

(j−1)pk−3

= lim

k→+∞

1 + 6

(j+ 1)(j−1)pk−3(j+ 1)

(j−1)pk−3

= exp 6

j+ 1

and

k→+∞lim

(j+ 1)pk−3 2(j+ 1)k−3

2pk

· 2

p 2pk

= lim

k→+∞

1 + 3p−6 2(j+ 1)pk−3p

2pk

LIWEI CHEN AND YUNUS E. ZEYTUNCU

= exp

3p−6 j+ 1

.

Therefore, since ^j+1_j−1 ≥

j+1 j

2

, we obtain

k→+∞lim

kB_λ(z^−jkz^k)k^p_λ

kz^−jkz^kk^p_λ ≈ lim

k→+∞

j j+ 1

(2jk−3)p j+ 1 j−1

(j−1)pk−3

p 2

2pk

≥ lim

k→+∞

j+ 1 j

2(j−1)pk−6−(2jk−3)p

p 2

2pk

= lim

k→+∞

j+ 1 j

3p−6−2pk

p 2

2pk

≥ lim

k→+∞

1 +1

j −2pk

p 2

2pk

=∞.

This shows that for any p > 2, the weighted Bergman projection B_λ is unbounded onL^p(D^∗, λ).

Now, we deduce the unboundedness of B^µ

H from unboundedness of the weighted Bergman projections as follows. LetBλ˜ be the weighted Bergman projection onL²_a(D^∗,λ), where ˜˜ λ(z) =|z|²exp

−_|z|¹

,z∈D^∗. Then by the inflation principle (see [Zey13, Che17]), if B^µ

H is bounded on L^p(H, µ) then B_λ˜ is bounded onL^p(D^∗,λ).˜

Note that the behavior of the sequence

z^−jkz^k _k∈

Z⁺ under the projec- tion B_λ˜ can be obtained from the behavior of the same sequence under the projection B_λ by shifting the indices of the integral I(α) by 2. More precisely, we have

kB_˜λ(z^−jkz^k)k^p_˜

λ

kz^−jkz^kk^p_˜

λ

= [I(−2jk+ 3)]^p·I(−(j+ 1)pk+ 3) [I(−2(j+ 1)k+ 3)]^p·I(−(j−1)pk+ 3) and

k→+∞lim

kB_λ˜(z^−jkz^k)k^p_˜

λ

kz^−jkz^kk^p_˜

λ

· j

j+ 1 2p

j+ 1 j−1

2

= lim

k→+∞

kB_λ(z^−jkz^k)k^p_λ kz^−jkz^kk^p_λ .

Hence, for anyp >2 the weighted Bergman projection B_˜λ is unbounded on L^p(D^∗,λ) and we conclude that˜ B^µ

H is bounded on L^p(H, µ) if and only ifp= 2.

3. Proof of Theorem 1.2

In this proof, we again study the weighted Bergman space L²_a(D^∗, λ), where λ(z) = exp

−_|z|¹

. Note that, if we denote the weighted Bergman kernel associated toL²_a(D^∗, λ) byB_λ(z, ζ), then we have

Bλ(z, ζ) =

+∞

X

k=−∞

1

I(2k+ 1)z^kζ^k. By (2) and (3), we see that

B_λ(z, z) =

−1

X

k=−∞

+

+∞

X

k=0

1

I(2k+ 1)|z|^2k

≈ |z|⁻² I(−1)+

+∞

X

k=2

|z|^−2k Γ(2k−2)+

+∞

X

k=0

(k+ 1)|z|^2k

≈ |z|⁻³sinh|z|⁻¹+ 1 (1− |z|²)². Let B^µ

H(z, ζ) be the Bergman kernel on H with respect to the weight µ(z₁, z₂) = exp

−_|z¹

1|

, and letB^µ0

H(z, ζ) be the Bergman kernel on Hwith respect to the weight µ⁰(z1, z2) = |z₁|^t, where t ∈ R. By looking at the biholomorphism

Φ :H→D^∗×D, where Φ(z₁, z₂) =

z₁,z₂ z₁

and by the biholomorphic equivalence of kernels, see [Che17, Corollary 2.4 and Lemma 3.1], we conclude that

B^µ0

H(z, ζ)

= detJ_CΦ(z)detJ_CΦ(ζ)B^µ0◦Φ−1

D^∗×D (Φ(z),Φ(ζ))

= 1 z1

· 1

ζ₁ ·B^µ0◦Φ−1

D^∗ (z1, ζ1)·B_D z₂

z1

,ζ₂ ζ1

= s

2 · 1

(z₁ζ₁)^k+2 + 1−s

2

· 1 (z₁ζ₁)^k+1

· 1

(1−z₁ζ₁)² · 1

1− ^z_z²

1 ·^ζ2

ζ₁

2

wheret=s+ 2k,k∈Zand s∈(0,2], and that B^µ

H(z, ζ) = detJ_CΦ(z)detJ_CΦ(ζ)B^µ◦Φ−1

D^∗×D(Φ(z),Φ(ζ))

= 1 z₁ · 1

ζ₁ ·B^µ◦Φ−1

D^∗ (z₁, ζ₁)·B_D z₂

z₁,ζ₂ ζ₁

LIWEI CHEN AND YUNUS E. ZEYTUNCU

=B_λ(z1, ζ1)· 1 z₁ · 1

ζ₁ · 1

1−^z_z²

1 ·^ζ2

ζ₁

2.

Therefore, on the diagonal line, we have B^µ0

H(z, z) = s

2 · 1

|z₁|^2k+4 + 1−s

2

· 1

|z₁|^2k+2

· 1

(1− |z₁|²)² · 1

1−^|z_|z^2|2

1|²

2

and B^µ

H(z, z)≈

|z₁|⁻³sinh|z₁|⁻¹+ 1 (1− |z₁|²)²

· 1

|z₁|² · 1

1−^|z_|z^2|2

1|²

2.

Recall that δ(z) =|z₁|, hence B^µ0

H(z, z)·δ(z)^τ

= s

2 · 1

|z₁|^2k+4 +

1− s 2

· 1

|z₁|^2k+2

· 1

(1− |z₁|²)² · 1

1−^|z_|z^2|2

1|²

2 · |z₁|^τ

≤ |z₁|^τ−(2k+4)· 1

(1− |z₁|²)² · 1

1−^|z_|z^2|2

1|²

2

≤C

for someτ ∈Rasz→0 inside any coneV_γ ={(z₁, z₂)∈C²: γ|z₂|<|z₁|}, whereγ >1. Whereas, for the exponential weightµon H,

B^µ

H(z, z)·δ(z)^τ

≈

|z₁|⁻³sinh|z₁|⁻¹+ 1 (1− |z₁|²)²

· 1

|z₁|² · 1

1−^|z_|z^2|2

1|²

2 · |z₁|^τ

≈ |z₁|^τ−5exp 1

|z₁|

· 1

1− ^|z_|z^2|2

1|²

2

is unbounded for all τ ∈Rasz→0 inside any coneVγ, whereγ >1.

Remark. If we correct the distance δ(z) by a factor of the weight µ(z), then we get

B^µ

H(z, z)·µ(z)·δ(z)^τ ≈ |z₁|^τ−5·

1−|z₂|²

|z₁|² −2

≤C forτ ≥5 as z→0 inside any cone Vγ, where γ >1.

4. Proof of Theorem 1.3

In view of the Forelli-Rudin inflation principle, see for example [Zey13, Proposition 4.4], unboundedness of B_H∞ can be deduced from unbounded- ness of the corresponding weighted Bergman projection on the punctured disc D^∗. Note that, in [Zey13], the smooth radial weights vanish at infinite order at the boundary of the unit disc. Hence, integration by parts plays a crucial role in obtaining asymptotics of the moment function of these weights. Whereas, here the radial weight λ(z) = exp

−_|z|¹

is vanishing of any order only at the origin, the nonsmooth boundary of the punctured discD^∗. Since we do not have vanishing conditions on the smooth boundary of the punctured disc, we do not base our argument on successive integra- tion by parts. Instead we use the asymptotic information from the second section.

Proof of Theorem 1.3. By [Zey13, Proposition 4.4], for p ∈ (1,∞) if the weighted Bergman projection B_λ is unbounded on the weighted space L^p(D^∗, λ), then the Bergman projectionB_H∞associated toH∞is unbounded onL^p(H∞). In Section 2, we have already showed thatB_λ is unbounded on L^p(D^∗, λ) forp >2, therefore we conclude thatB_H∞ is bounded onL^p(H∞)

if and only if p= 2.

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(Liwei Chen) The Ohio State University, Department of Mathematics, Colum- bus, OH 43210

[email protected]

(Yunus E. Zeytuncu)University of Michigan - Dearborn, Department of Math- ematics and Statistics, Dearborn, MI 48128

[email protected]

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