Interpolations of harmonic Bergman functions on smooth bounded domains

Interpolations of harmonic Bergman functions on smooth bounded domains

Kiyoki Tanaka

Osaka City University Advanced Mathematical Institute

December, 22, 2013 / 22th Functional Space Seminar

Outline

1 Setting

2 Results

3 Outline of the proof

Bergman space (origin)

LetD⊂Cbe the unit disc.

L²_a(D,dA) ={f : analytic onD,kfk₂<∞}

f ∈L²_a(D,dA)has the following formula:

f(z) = Z

D

f(w) 1

π(1−zw)¯ ²dA(w)forz ∈D K(z,w) = _π(1−z¹_{w)¯ 2} : Bergman kernel

Forf ∈L²₍D,dA),Pf(z) :=R

Df(w)_π(1−z¹_{w)¯ 2}dA(w): Bergman projection

Bergman space (origin)

LetD⊂Cbe the unit disc.

L²_a(D,dA) ={f : analytic onD,kfk₂<∞}

f ∈L²_a(D,dA)has the following formula:

f(z) = Z

D

f(w) 1

π(1−zw)¯ ²dA(w)forz ∈D K(z,w) = _π(1−z¹_{w)¯ 2} : Bergman kernel

Forf ∈L²₍D,dA),Pf(z) :=R

Df(w)_π(1−z¹_{w)¯ 2}dA(w): Bergman projection

Bergman space (origin)

LetD⊂Cbe the unit disc.

L²_a(D,dA) ={f : analytic onD,kfk₂<∞}

f ∈L²_a(D,dA)has the following formula:

f(z) = Z

D

f(w) 1

π(1−zw)¯ ²dA(w)forz ∈D K(z,w) = _π(1−z¹_{w)¯ 2} : Bergman kernel

Forf ∈L²₍D,dA),Pf(z) :=R

Df(w)_π(1−z¹_{w)¯ 2}dA(w): Bergman projection

Examples of research

Characterization of operators onL²_a(D)( Toeplitz operator, Hankel operator, composition operator, etc. )

Change setting:

measuredA weighted measureϕ(w)dA(w)(ϕ(w) = (1− |w|²)^α )

D certain domain ( upper-half plane, ball inCⁿ, pseudo-convex domain)

analytic solution of certain differential equation ( harmonic, heat equation )

Examples of research

Characterization of operators onL²_a(D)( Toeplitz operator, Hankel operator, composition operator, etc. )

Change setting:

measuredA weighted measureϕ(w)dA(w)(ϕ(w) = (1− |w|²)^α )

D certain domain ( upper-half plane, ball inCⁿ, pseudo-convex domain)

analytic solution of certain differential equation ( harmonic, heat equation )

Harmonic Bergman space

1≤p<∞,Ω⊂Rⁿ: bounded smooth domain.

b^p(Ω) :={f : harmonic inΩandkfk_p<∞}

b^p: harmonic Bergman space.

b^p(Ω)⊂L^p(Ω): closed subspace f ∈b²(Ω)has a reproducing property:

f(x) = Z

Ω

R(x,y)f(y)dy forx ∈Ω R(·,·):harmonic Bergman kernel.

Example of the harmonic Bergman kernel

WhenΩ =B(unit ball),

R_B(x,y) = (n−4)|x|⁴|y|⁴+ (8x ·y−2n−4)|x|²|y|²+n n|B|((1− |x|²)(1− |y|²) +|x−y|²)¹⁺ⁿ² and

R_B(x,x) = (n−4)|x|⁴+2n|x|²+n n|B|(1− |x|²)ⁿ

Preparation for results

Lemma (covering lemma)

Let0< δ < ¹₄. We can choose N (independ ofδ),{λ_i} ⊂Ωand disjoint covering{E_i}forΩ.

1 E_i is measurable set for any i ∈N;

2 E_i ⊂B(λ_i, δr(λi))for any i ∈N;

3 {B(λ_i,3δr(λ_i))}is uniformly finite intersection with bound N

Representation theorem

Theorem (T. (2012))

Let1<p<∞,Ωbe a bounded smooth domain. There exists0< δ0

such that if{λ_i}satisfies covering lemma forδ < δ₀, then

A_p,{λi} :`^p→b^p is a bounded onto map, where the operator A_p,{λi} is defined by

A_p,{λi}{a_i}(x) :=

∞

X

i=1

a_iR(x, λ_i)r(λ_i)^(1−1p)n,

where r(x)denotes the distance between x and∂Ω.

Fix a defining functionηforΩs.t.|∇η|²=1+ηωfor someω∈C^∞( ¯Ω).

B.R. Choe, H. Koo and H. Yi (2004) introduced the following kernel R₁(x,y) :=R(x,y)−1

2∆y(η²(y)R_x(y)).

and shown thatR₁has the reproducing property.

Theorem (T. (2013))

Let1≤p<∞andΩbe a smooth bounded domain. Then, we can choose a sequence{λ_i}inΩsuch that A₁:`^p→b^pis a bounded onto map, where the operator A₁is defined by

A₁{a_i}(x) :=

∞

X

i=1

a_iR₁(x, λ_i)r(λ_i)^(1−1p)n,

Interpolating of harmonic Bergman functions

Let denoteV :b^p(Ω)→l^p by

V_p,{λi}f :={r(λi)

n pf(λi)}.

It is known thatA^∗_p,{λ

i}=V_q,{λi} for 1<p<∞,q: ¹_p+_q¹ =1.

Theorem (T. (2013))

Let1<p<∞. We can choose a positive constantρ0satisfying the following condition;

if{λ_i}_i ⊂Ωsatisfy quasi-hyperbolic distanceρ(λ_i, λ_j)> ρ₀for i 6=j, then V :b^p(Ω)→l^pis bounded and onto.

Interpolating of harmonic Bergman functions

Let denoteV :b^p(Ω)→l^p by

V_p,{λi}f :={r(λi)

n pf(λi)}.

It is known thatA^∗_p,{λ

i}=V_q,{λi} for 1<p<∞,q: ¹_p+_q¹ =1.

Theorem (T. (2013))

Let1<p<∞. We can choose a positive constantρ0satisfying the following condition;

if{λ_i}_i ⊂Ωsatisfy quasi-hyperbolic distanceρ(λ_i, λ_j)> ρ₀for i 6=j, then V :b^p(Ω)→l^pis bounded and onto.

ρ(x,y) := inf

γ∈Γx,y

Z

γ

1

r(z)ds(z)

The preceding works

R.R. Coifman - R. Rochberg (1980) : holomorphic on unit ball in Cⁿ

B.R. Choe - H. Yi (1998) : harmonic on upper half space inRⁿ K. Tanaka : harmonic on smooth bounded domain

The preceding result

Theorem (Kang-Koo 2002)

LetΩbe a smooth bounded domain andαandβbe multi-indices.

Then, there exist C_α,β >0and C>0such that for any x,y ∈Ω

|D_x^αD^β_yR(x,y)| ≤ C_α,β d(x,y)^n+|α|+|β|

and

R(x,x)≥ C r(x)ⁿ

where d(x,y) :=r(x) +r(y) +|x−y|and r(x)is the distance between x and boundary ofΩ.

Outline of the proof of representation theorem

Lemma (covering lemma)

Let0< δ < ¹₄. We can choose N (independ ofδ),{λ_i} ⊂Ωand disjoint covering{E_i}forΩ.

1 E_i is measurable set for any i ∈N;

2 E_i ⊂B(λ_i, δr(λi))for any i ∈N;

3 {B(λ_i,3δr(λ_i))}is uniformly finite intersection with bound N Remark.

If there exists a positive constantc >0 such that{B(λ_i,cr(λ_i))}is uniformly finite intersection, thenA_p,{λi} andV_p,{λi} are bounded for

∞.

We define the operatorsU_p,{λi} :b^p→`^pandS_p,{λi}:b^p→b^pas following;

S_p,{λi}f(x) :=

∞

X

i=1

R(x, λ_i)f(λ_i)|E_i|

U_p,{λi}(f) :={|E_i|f(λi)r(λi)⁻⁽¹⁻

1 p)n

}_i

where{E_i}_i is the disjoint covering ofΩsuch thatλi ∈E_i for anyi ∈N. BecauseS=A◦U, we may show thatSis bijective map. By

calculatingkS−Idk, we can give the condition thatSis bijective.

Interpolating of harmonic Bergman functions (again)

Let denoteV :b^p(Ω)→l^p by

V_p,{λi}f :={r(λi)

n pf(λi)}.

It is known thatA^∗_p,{λ

i}=V_q,{λi} for 1<p<∞,q: ¹_p+_q¹ =1.

Theorem (T. (2013))

Let1<p<∞. We can choose a positive constantρ0satisfying the following condition;

if{λ_i}_i ⊂Ωsatisfy quasi-hyperbolic distanceρ(λ_i, λ_j)> ρ₀for i 6=j, then V :b^p(Ω)→l^pis bounded and onto.

Outline of the proof of interpolating sequence theorem

We assumeρ(λ_i, λ_j)> ρ >1.

Then, there existδ >0 such that{B(λ_i, δr(λi))}is disjoint.We put W_p,{λi}{a_i}:=V ◦A{a_i}={r(λ_j)^np

∞

X

i=1

a_iR(λ_j, λ_i)r(λ_i)^(1−1p)n}_j.

We choose the{λ_i}_i such that the operatorW is bijective. We put diagonal part ofW

D{a_i}:={a_jR(λ_j, λ_j)r(λ_j)ⁿ} and off diagonal part ofW

E{a_i}:={r(λ_j)^npX

i6=j

a_iR(λ_j, λ_i)r(λ_i)^(1−1p)n}_j.

Outline of the proof of interpolating sequence theorem

We assumeρ(λ_i, λ_j)> ρ >1.

Then, there existδ >0 such that{B(λ_i, δr(λi))}is disjoint. We put W_p,{λi}{a_i}:=V ◦A{a_i}={r(λ_j)^np

∞

X

i=1

a_iR(λ_j, λ_i)r(λ_i)^(1−1p)n}_j.

We choose the{λ_i}_i such that the operatorW is bijective. We put diagonal part ofW

D{a_i}:={a_jR(λ_j, λ_j)r(λ_j)ⁿ} and off diagonal part ofW

nX ₍₁₋¹_)n

Outline of the proof of interpolating sequence theorem

We assumeρ(λ_i, λ_j)> ρ >1.

Then, there existδ >0 such that{B(λ_i, δr(λi))}is disjoint. We put W_p,{λi}{a_i}:=V ◦A{a_i}={r(λ_j)^np

∞

X

i=1

a_iR(λ_j, λ_i)r(λ_i)^(1−1p)n}_j.

We choose the{λ_i}_i such that the operatorW is bijective. We put diagonal part ofW

D{a_i}:={a_jR(λ_j, λ_j)r(λ_j)ⁿ} and off diagonal part ofW

E{a_i}:={r(λ_j)^npX

i6=j

a_iR(λ_j, λ_i)r(λ_i)^(1−1p)n}_j.

BecauseD: bijective andW =D+E,kEk< _kD¹−1k ⇒W : bijective.

Moreover, sinceR(x,x)r(x)ⁿ≈1,kDk ≈1. We have only to give the estimate forkEk. By using Hölder’s inequality, we have

kE{a_i}k_l^p .δ

−n(p−1) p

X^∞

i=1

|a_i|^pr(λi)

1

qX

j6=i

r(λj)ⁿ⁻

1

q|R(λ_j, λi)|¹_p

We calculate a part of summation;

r(λ_i)^1qX

j6=i

r(λ_j)^n−1q|R(λ_j, λ_i)|. Z

Ω\E_δ(λi)

r(λ_i)^1qr(z)^−1q d(z, λ_i)ⁿ dz

BecauseD: bijective andW =D+E,kEk< _kD¹−1k ⇒W : bijective.

Moreover, sinceR(x,x)r(x)ⁿ≈1,kDk ≈1. We have only to give the estimate forkEk. By using Hölder’s inequality, we have

kE{a_i}k_l^p .δ

−n(p−1) p

X^∞

i=1

|a_i|^pr(λi)

1

qX

j6=i

r(λj)ⁿ⁻

1

q|R(λ_j, λi)|¹_p

We calculate a part of summation;

r(λ_i)^1qX

j6=i

r(λ_j)^n−1q|R(λ_j, λ_i)|. Z

Ω\E_δ(λi)

r(λ_i)^1qr(z)^−1q d(z, λ_i)ⁿ dz

BecauseD: bijective andW =D+E,kEk< _kD¹−1k ⇒W : bijective.

Moreover, sinceR(x,x)r(x)ⁿ≈1,kDk ≈1. We have only to give the estimate forkEk. By using Hölder’s inequality, we have

kE{a_i}k_l^p .δ

−n(p−1) p

X^∞

i=1

|a_i|^pr(λi)

1

qX

j6=i

r(λj)ⁿ⁻

1

q|R(λ_j, λi)|¹_p

We calculate a part of summation;

r(λ_i)^1qX

j6=i

r(λ_j)^n−1q|R(λ_j, λ_i)|. Z

Ω\E_δ(λi)

r(λ_i)^1qr(z)^−1q d(z, λ_i)ⁿ dz

BecauseD: bijective andW =D+E,kEk< _kD¹−1k ⇒W : bijective.

Moreover, sinceR(x,x)r(x)ⁿ≈1,kDk ≈1. We have only to give the estimate forkEk. By using Hölder’s inequality, we have

kE{a_i}k_l^p .δ

−n(p−1) p

X^∞

i=1

|a_i|^pr(λi)

1

qX

j6=i

r(λj)ⁿ⁻

1

q|R(λ_j, λi)|¹_p

We calculate a part of summation;

r(λ_i)^1qX

j6=i

r(λ_j)^n−1q|R(λ_j, λ_i)|. Z

Ω\E_δ(λi)

r(λ_i)^1qr(z)^−1q d(z, λ_i)ⁿ dz

Lemma (Pseudo-hyperbolic distance)

There exist constants C₁>0and C₂>0such that for any x,y ∈Ω 1

d(x,y) ≤ exp(−^ρ(x,y)−C_C ¹

2 )

min{r(x),r(y)}.

By using quasi-hyperbolic distance conditionρ(λ_i, λ_j)> ρ, for 0< <1 we have

Z

Ω\E_δ(λ_i)

r(λ_i)^1qr(z)^−1q

d(z, λi)ⁿ dz .exp(−(ρ−C₁ C₂ ))→0 asρ→ ∞.

Then, there existsρ₀>0 such thatkEk< _kD¹−1k ifρ(λ_i, λ_j)> ρ₀(i6=j).

Lemma (Pseudo-hyperbolic distance)

There exist constants C₁>0and C₂>0such that for any x,y ∈Ω 1

d(x,y) ≤ exp(−^ρ(x,y)−C_C ¹

2 )

min{r(x),r(y)}.

By using quasi-hyperbolic distance conditionρ(λ_i, λ_j)> ρ, for 0< <1 we have

Z

Ω\E_δ(λ_i)

r(λ_i)^1qr(z)^−1q

d(z, λi)ⁿ dz .exp(−(ρ−C₁ C₂ ))→0 asρ→ ∞.

Then, there existsρ₀>0 such thatkEk< _kD¹−1k ifρ(λ_i, λ_j)> ρ₀(i6=j).

References

[1] B. R. Choe, H. Koo and H. Yi,Projections for harmonic Bergman spaces and applications, J. Funct. Anal.,216(2004), 388–421.

[2] B. R. Choe and H. Yi,Representations and interpolations of harmonic Bergman functions on half-spaces, Nagoya Math. J.151(1998), 51–89.

[3] R.R. Coifman and R. Rochberg,Representation Theorems for

Holomorphic and Harmonic functions in L^p,Ast´erisque77(1980), 11–66.

[4] C. Fefferman,The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Inv. Math. 26(1974), 1–65.

[5] H. Kang and H. Koo,Estimates of the harmonic Bergman kernel on smooth domains, J. Funct. Anal.,185(2001), 220–239.

[6] K. Tanaka,Atomic decomposition of harmonic Bergman functions, Hiroshima Math. J., 42 (2012), 143–160.

[7] K. Tanaka,Interpolation theorem for harmonic Bergman functions, RIMS Kôkyûroku Bessatu B43 (2013), 183–191.