WEIGHTED ESTIMATES FOR SOLUTIONS OF THE ∂ -EQUATION FOR LINEALLY CONVEX DOMAINS OF FINITE TYPE AND APPLICATIONS TO WEIGHTED BERGMAN PROJECTIONS

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WEIGHTED ESTIMATES FOR SOLUTIONS OF THE

∂ -EQUATION FOR LINEALLY CONVEX DOMAINS OF FINITE TYPE AND APPLICATIONS TO

WEIGHTED BERGMAN PROJECTIONS

Philippe Charpentier, Y Dupain, M Mounkaila

To cite this version:

Philippe Charpentier, Y Dupain, M Mounkaila. WEIGHTED ESTIMATES FOR SOLUTIONS OF THE ∂ -EQUATION FOR LINEALLY CONVEX DOMAINS OF FINITE TYPE AND APPLICA- TIONS TO WEIGHTED BERGMAN PROJECTIONS. 2016. �hal-01311050�

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LINEALLY CONVEX DOMAINS OF FINITE TYPE AND APPLICATIONS TO WEIGHTED BERGMAN PROJECTIONS

P. CHARPENTIER, Y. DUPAIN & M. MOUNKAILA

ABSTRACT. In this paper we obtain sharp weighted estimates for solutions of the∂- equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the formL^p(Ω,δ^γ),δbeing the distance to the boundary, with gain on the index pand the exponentγ. These estimates allow us to extend theL^p(Ω,δ^γ)and lipschitz regularity results for weighted Bergman projection obtained in [CDM14b] for convex domains to more general weights.

1. INTRODUCTION

The study of the regularity of the Bergman projection onto holomorphic functions in a given Hilbert space is a very classical subject. When the Hilbert space is the standard LebesgueL²space on a smoothly bounded pseudoconvex domainΩinCⁿ, many results are known and there is a very large bibliography.

When the Hilbert space is a weightedL²space on a smoothly bounded pseudoconvex domainΩinCⁿ, it is well known for a long time that the regularity of the Bergman projection depends strongly on the weight ([Koh73], [Bar92], [Chr96]). Until last years few results where known (see [FR75], [Lig89], [BG95], [CL97]) but recently some positive and negative results where obtained by several authors (see for example [Zey11], [Zey12], [Zey13b], [Zey13a], [CDM14b], [CDM15], [vZ], [Zey16] and references therein).

In this paper we are interested in some generalization of the result obtained in [CDM14b]

for convex domains of finite type.

LetΩbe a convex domain of finite type inCⁿ. Letgbe a gauge function forΩand define ρ0=g⁴e¹⁻¹^/^g−1. LetP_ω₀ be the Bergman projection of the spaceL²(Ω,ω0), whereω0= (−ρ0)^r,r∈Q₊. Then in [CDM14b, Theorem 2.1] we proved thatP_ω₀ maps continuously the spacesL^p

Ω,δ_Ω^β,p∈]1,+∞[, 0<β+1≤p(r+1), into themselves,δ_Ω being the distance to the boundary of Ω. Here we consider a weightω which is a non negative rational power of a C² function inΩequivalent to the distance to the boundary and we prove that the Bergman projectionP_ω of the Hilbert spaceL²(Ω,ω)maps continuously the spacesL^p

Ω,δ_{∂ Ω}^β ,p∈]1,+∞[, 0<β+1≤r+1 into themselves and the lipschitz spacesΛ_α(Ω), 0<α≤¹/^m, into themselves.

This result is obtained comparing the operatorsP_ω₀ andP_ω with the method described in [CDM15]. To do it we need to have weightedL^p Ω,δ_Ω^γestimates with appropriate gains on the indexpand on the powerγfor solution of the∂-equation.

This is done, with sharp estimates, for a general lineally convex domain of finite type using the method introduced in [CDM14a], which overcomes the fact that the Diederich- Fornaess support function is only locally defined and that it is not possible do make a division with good estimates in non convex domains.

2010Mathematics Subject Classification. 32T25, 32T27.

Key words and phrases. lineally convex, finite type,∂-equation, weighted Bergman projection.

1

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Our results extend the results (without weights) obtained for convex domains of finite type by A. Cumenge in [Cum01a] and [Cum01b] and B. Fisher in [Fis01] (see also T. Hefer [Hef02]).

2. NOTATIONS AND MAIN RESULTS

Throughout this paper we will use the following general notations:

• Ω is a smoothly bounded lineally convex domain of finite type m in Cⁿ (see [CDM14a] for a precise definition).

• ρ is a smooth defining function ofΩsuch that, forδ0sufficiently small, the do- mainsΩt={ρ(z)<t},−δ0≤t≤δ0, are all lineally convex of finite typem.

• δ_Ωdenotes the distance to the boundary ofΩ.

• For any real numberγ>−1, we denote byL^p Ω,δ_Ω^γtheL^p-space onΩfor the measureδ_Ω^γ(z)dλ(z),λbeing the Lebesgue measure.

Our first results give sharpL^q

Ω,δ_Ω^γ^′estimates for solutions of the∂-equation inΩ with data inL^p Ω,δ_Ω^γ:

Theorem 2.1. Let N be a positive large integer. letγandγ^′be two real numbers such that max{−1,γ−¹/m} ≤γ^′≤γ≤N−2. Then there exists a linear operator T , depending onρ and N, such that, for any∂-closed(0,r)-form with coefficients in L^p Ω,δ_Ω^γ, p∈[1,+∞], T f is a solution of the equation∂(T f) =f satisfying the following estimate:

(1) If1≤p<^m(γ_{1−m(γ−γ}^′⁺ⁿ⁾⁺²′), T maps continuously the space of∂-closed forms with coefficients in L^p Ω,δ_Ω^γinto the space of forms whose coefficients are in L^q

Ω,δ_Ω^γ^′ with¹_q=¹_p−^{1−m(γ−γ}_m(γ′+n)+2^′⁾;

(2) If p=m(γ+n) +2, T maps continuously the space of∂-closed forms with coefficients in L^p Ω,δ_Ω^γinto the space of forms whose coefficients are in BMO(Ω);

(3) If p∈]m(γ+n) +2,+∞], then T maps continuously the space of∂-closed forms with coefficients in L^p Ω,δ_Ω^γinto the space of forms whose coefficients are in the lipschitz spaceΛα(Ω)withα=_m¹h

1−^m(γ+n)+2_p i .

Note that, ifγ^′<γ, then^m(γ_{1−m(γ−γ}^′⁺ⁿ⁾⁺²′)>m(γ+n)+2, and (3) is sharper than (1). Moreover, without weights, these estimates are known to be sharp (see [CKM93]).

The two next propositions, which are immediate corollaries of the theorem, will be used in the last section:

Proposition 2.1. There exists a constantε0>0such that, for all large integer N and all

−1<γ≤N−2, there exists a linear operator T solving the∂-equation inΩ such that, for all p∈[1,+∞[, there exists a constant CN,p>0such that for all∂-closed(0,r)-form

f ,1≤r≤n−1, onΩ, we have Z

Ω|T f|^pδ_Ω^γdλ ≤C_N,p Z

Ω|f|^pδ_Ω^γ+ε⁰dλ.

Proposition 2.2. There exist a linear operator T solving the∂-equation inΩand a con- stantε0>0such that, for all−1<γ≤N−2and all p∈[1,+∞[, there exists a constant C_N,p>0such that for all∂-closed(0,r)-form f ,1≤r≤n−1, we have

Z

Ω|T f|^p+ε⁰δ_Ω^γdλ ≤C_N,p Z

Ω|f|^pδ_Ω^γdλ.

Our last estimate for solutions of the∂-equation is a generalization to lineally convex domains of an estimate obtained by A. Cumenge ([Cum01b]) for convex domains of finite type:

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Theorem 2.2. For allα >0 there exists a constant C>0 such that, for all smooth∂- closed(0,r)-form f ,1≤r≤n−1, onΩ, there exists a solution of the equation∂u=f , continuous onΩsuch that

Z

Ω|u|δ_Ω^α−1dλ≤C1 α

Z

Ωkfk_kδ_Ω^αdλ,

where the normkfk_kwas introduced in[BCD98](see[CDM14a]for details, the definition is recalled in section 3.2).

Note that the estimate given by Theorem 2.1 when p=q=1 (and thenγ^′=γ−¹/m) is weaker than the one given above.

An immediate application of this last estimate is the characterization of the zero sets of the weighted Nevanlinna classes (called Nevanlinna-Djrbachian classes in [Cum01b]) obtained by A. Cumenge for convex domains:

Theorem 2.3. A divisor D inΩ can be defined by a holomorphic function satisfying R

Ωln⁺|f|δ_Ω^α−1dλ<+∞,α>0, if and only if it satisfy the generalized Blaschke condition R

Dδ^α+1dλ2n−2<+∞.

As the proof of such result using Theorem 2.2 is very classical we will not give any detail on it in this paper.

The two propositions 2.1 and 2.2 will be used to generalize some estimates obtained for weighted Bergman projections of convex domains of finite type in [CDM14b]:

Theorem 2.4. Let D be smoothly bounded convex domain of finite type inCⁿ. Letχ be anyC²non negative function in D which is equivalent to the distanceδDto the boundary of D and letηbe a strictly positiveC¹function on D. Let P_ω be the (weighted) Bergman projection of the Hilbert space L²(D,ω)whereω=η χ^r with r a non negative rational number.Then:

(1) For p∈]1,+∞[and−1<β≤r, P_ωmaps continuously L^p

D,δ_D^βinto itself.

(2) For0<α≤¹/mP_ωmaps continuously the Lipschitz spaceΛα(D)into itself.

This theorem combined with Theorem 2.1 extends to weighted situations the Corollary 1.3 of [Cum01a]

Corollary. Under the assumptions of Theorem 2.4, the solution of the equation∂u= f which is orthogonal to holomorphic functions in L²(D,ω) where f is a (0,1)-form ∂- closed with coefficients in L^p(Ω,δ_Ω^γ),−1<γ, belongs to:

(1) L^q(D,δ_D^γ^′), with ¹_q=¹_p−^{1−m(γ−γ}_m(γ′+n)+2^′⁾ andmax{−1,γ−¹/^m}<γ^′≤γ, ifγ^′≤r,1≤ p<^m(γ_{1−m(γ−γ}^′⁺ⁿ⁾⁺²′), and q>1;

(2) Λα(D), withα=_m¹h

1−^m(γ+n)+2_p i

, if p∈]m(γ+n) +2,+∞].

3. PROOFS OF THEOREMS2.1AND2.2

First of all by standard regularization procedure, it suffices to prove theorems 2.1, and 2.2 for forms smooth inΩ.

To solve the∂-equation on a lineally convex domain of finite type, we use the method introduced in [CDM14a]. We now briefly recall the notations and main results from that work.

If f is a smooth(0,r)-form∂-closed, the following formula was established f(z) = (−1)^q+1∂z

Z Ω

f(ζ)∧K_N¹(z,ζ)

− Z

Ω

f(ζ)∧P_N(z,ζ),

whereK_N¹ (resp. P_N) is the component of a kernelK_N (formula (2.7) of [CDM14a]) of bi-degree(0,r)inzand(n,n−r−1)inζ (resp.(0,r)inzand(n,n−r)inζ) constructed

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with the method of [AB82] using the Diederich-Fornaess support function constructed in [DF03] (see also Theorem 2.2 of [CDM14a]) and the functionG(ξ) = _ξ¹N with a sufficiently large numberN(instead ofG(ξ) =_ξ¹ in formula (2.7) of [CDM14a]).

Then, the form^R_Ω f(ζ)∧P_N(z,ζ)is∂-closed and the operatorT solving the∂-equation in theorems 2.1 and 2.2 is defined on smooth forms by

T f(z) = Z

Ωf(ζ)∧K_N¹(z,ζ)−∂^∗^N Z

Ωf(ζ)∧PN(z,ζ)

,

where∂^∗^N is the canonical solution of the∂-equation derived from the theory of the

∂-Neumann problem on pseudoconvex domains of finite type.

This formula is justified by the fact that, when the coefficients of f are inL¹ Ω,δ_Ω^γ (γ>−1) then, given a large integers, ifNis chosen sufficiently large, the coefficients of the form^R_Ωf(ζ)∧P_N(z,ζ)are in the Sobolev spaceL²_s(Ω). More precisely, it is clear that lemmas 2.2 and 2.3 of [CDM14a] remains true with weighted estimates depending on the choice ofN:

Lemma 3.1. For r≥1 andγ≤N, all the z-derivatives of P_N(z,ζ) (−ρ(ζ))^−γ are uni- formly bounded inΩ×Ω, and, for each positive integer s, there exists a constant Cs,N,γ

such that, if f is(0,r)-form with coefficients in L¹(Ω,δ_Ω^γ),

Z

Ωf(ζ)∧P_N(z,ζ) _L₂

s(Ω)

≤C_s,N,γkfk_L1(Ω,δ_Ω^γ).

AsΩis assumed to be smooth and of finite type, the regularity results of the∂-Neumann problem ([KN65] and [Cat87])

Lemma 3.2. For r≥1 and−1<γ≤N, for each positive integer s, if f is a∂-closed (0,r)-form with coefficients in L¹(Ω,δ_Ω^γ)and g=^R_Ω f(ζ)∧PN(z,ζ), then∂^∗^N (g)is a solution of the equation∂u=g satisfying

∂^∗^N (g)

L²_s(Ω)≤C_s,N,γkfk_L1(Ω,δ^γ). Applying Sobolev lemma we immediately get:

Lemma 3.3. For r≥1, p∈[1,+∞]and−1<γ≤N, if f is a∂-closed(0,r)-form with coefficients in L¹(Ω,δ_Ω^γ)and g=^R_Ωf(ζ)∧P_N(z,ζ), then∂^∗^N (g)is a solution of the equation∂u=g satisfying

∂^∗^N(g)_C₁_(Ω)≤Ckfk_L1(Ω,δ^γ).

Finally the proofs of our theorems are reduced to the proofs of good estimates for the operatorT_K defined by

(3.1) T_K: f 7→

Z

Ωf(ζ)∧K_N¹(z,ζ).

To do it with some details we need to recall the anisotropic geometry ofΩand the basic estimates given in [CDM14a].

Forζ close to∂Ωandε≤ε0,ε0small, define, for all unitary vectorv, τ(ζ,v,ε) =sup{csuch that|ρ(ζ+λv)−ρ(ζ)|<ε,∀λ ∈C,|λ|<c}.

Letζ andεbe fixed. Then, an orthonormal basis(v1,v2, . . . ,vn)is called(ζ,ε)-extremal (orε-extremal, or simplyextremal) ifv1is the complex normal (toρ) atζ, and, fori>

1, vibelongs to the orthogonal space of the vector space generated by(v1, . . . ,vi−1)and minimizesτ(ζ,v,ε)in that space. In association to an extremal basis, we denote

τ(ζ,v_i,ε) =τi(ζ,ε).

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Then we defined polydiscsAPε(ζ)by AP_ε(ζ) =

( z=ζ+

n k=1∑

λkv_ksuch that |λk| ≤c₀Aτk(ζ,ε) )

,

c₀being sufficiently small, depending onΩ,P_ε(ζ)being the corresponding polydisc with A=1 and we also define

d(ζ,z) =inf{εsuch thatz∈P_ε(ζ)}.

The fundamental result here is thatdis a pseudo-distance which means that,∀α>0, there exist constantsc(α)andC(α)such that

(3.2) c(α)Pε(ζ)⊂P_αε(ζ)⊂C(α)Pε(ζ)andP_c(α)ε(ζ)⊂αP_ε(ζ)⊂P_C(α)ε(ζ).

Forζclose to∂Ωandε>0 small, the basic properties of this geometry are (see [Con02]

and [CDM14a]):

(1) Letw= (w1, . . . ,w_n)be an orthonormal system of coordinates centered atζ. Then

∂^|α+β^|ρ(ζ)

∂w^α∂w¯^β

. ε

∏iτ(ζ,wi,ε)^αⁱ^+βⁱ,|α+β| ≥1.

(2) Letνbe a unit vector. Leta^ν_αβ(ζ) =_{∂ λ}^∂^α_α⁺_∂^β_λ_¯^ρ_β(ζ+λ ν)_|λ=0. Then

1≤|α+β∑|≤2m

a^ν_αβ(ζ)τ(ζ,ν,ε)^α+β≃ε.

(3) If(v1, . . . ,v_n)is a(ζ,ε)-extremal basis andγ=∑ⁿ1a_jv_j6=0, then 1

τ(ζ,γ,ε) ≃

n

∑j=1

a_j τj(ζ,ε). (4) Ifvis a unit vector then:

(a) z=ζ+λv∈P_ε(ζ)implies|λ|.τ(ζ,v,ε), (b) z=ζ+λvwith|λ| ≤τ(ζ,v,ε)impliesz∈CP_ε(ζ).

(5) Ifνis the unit complex normal, thenτ(ζ,v,ε) =εand ifvis any unit vector and λ≥1,

(3.3) λ¹^/^mτj(ζ,v,ε).τj(ζ,v,λ ε).λ τj(ζ,v,ε), wheremis the type ofΩ.

Lemma 3.4. For z close to ∂Ω, ε small andζ ∈P_ε(z), in the coordinate system (ζi) associated to the(z,ε)-extremal basis, we have:

(1)

^{∂ ρ}∂ ζ_i(ζ)._τ ^ε

i(z,ε) (property (1) of the geometry recalled above);

(2) τ_i(ζ,ε)≃τ_i(z,ε)if c₀is chosen sufficiently small.

We now recall the detailed expression ofK_N¹ ([CDM14a] sections 2.2 and 2.3):

K_N¹(z,ζ) =

n−1 k=n−r∑

C^′_k

ρ(ζ)^k+Ns∧

∂ζ¯Qn−r

∧(∂z¯Q)^k+r−n∧(∂z¯s)^n−k−1

|z−ζ|^2(n−k)

1

K₀S(z,ζ) +ρ(ζ)k+N , where

s(z,ζ) =

n i=1∑

ζi−z_i

d(ζi−z_i) and

Q(z,ζ) = 1 K₀ρ(ζ)

n i=1∑

Q_i(z,ζ)d(ζi−z_i)

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with

S(z,ζ) =χ(z,ζ)S0(z,ζ)−(1−χ(z,ζ))|z−ζ|²=

n i=1∑

Qi(z,ζ) (zi−ζi),

S0being the holomorphic support function of Diederich-Fornaess (see [DF03] or Theorem 2.2 of [CDM14a]) andχa truncating function which is equal to 1 when both|z−ζ|and δ_Ω(ζ)are small and 0 if one of these expressions is large (see the beginning of Section 2.2 of [CDM14a] for a precise definition). Recall thatK₀is chosen so that

ℜe

ρ(ζ) + 1 K₀S(z,ζ)

<ρ(ζ) 2 , that is

(3.4)

ρ(ζ) + 1 K₀S(z,ζ)

&|ρ(ζ)|.

The following estimates of the expressions appearing inK_N¹ are basic (see [CDM14a]):

Lemma 3.5. Forζ∈P2ε(z)\Pε(z), we have:

ρ(ζ) + 1 K₀S(z,ζ)

&ε,(z,ζ)∈Ω¯ ×Ω.¯

Lemma 3.6. For z close to ∂Ω, ε small andζ ∈Pε(z), in the coordinate system (ζi) associated to the(z,ε)-extremal basis, we have:

(1) |Qi(z,ζ)|+|Qi(ζ,z)|._τ ^ε

i(z,ε) (see[DF06]);

(2) ^∂Q^{∂ ζ}ⁱ^(z,ζj ⁾

._τ ^ε

i(z,ε)τ_j(z,ε) (see[DF06]);

(3) ^∂

2Qi(z,ζ)

∂ ζ_j∂zk

+

^∂

2Qi(z,ζ)

∂ ζ_j∂zk

._τ ^ε

i(z,ζ)τj(z,ζ)τk(z,ζ) (see[DF06]).

To simplify notations, we will now do the proofs of the theorems only for(0,1)-forms, the general case of(0,r)-forms being identical except for complications in the notations.

The preceding lemmas and the properties of the geometry easily give the following estimates of the kernelK_N¹(for(0,1)-forms):

Lemma 3.7. Forεsmall enough and z sufficiently close to the boundary we have:

Ifζ ∈P_ε(z),

K_N¹(z,ζ). ρ(ζ)^N−1(|ρ(ζ)|+ε)εⁿ⁻¹

∏ⁿ⁻¹_i=1τi(z,ε)K¹₀S(z,ζ) +ρ(ζ)

N+n−1

1

|z−ζ|.

In particular:

Lemma 3.8. Forεsmall enough and z sufficiently close to the boundary:

(1) Ifε≤δ_{∂ Ω}(z), forζ ∈P_ε(z),

K_N¹(z,ζ). 1

∏ⁿ⁻¹_i=1τi(z,ε) 1

|z−ζ|. (2) Ifζ∈P_2ε(z)\P_ε(z)or z∈P_2ε(ζ)\P_ε(ζ)and k≤N+n−1,

K_N¹(z,ζ).|ρ(ζ)|^k ε^k

1

∏ⁿ⁻¹_i=1τi

1

|z−ζ|, and

∇_zK_N¹(z,ζ).|ρ(ζ)|^k ε^k+1

1

∏ⁿ⁻¹_i=1τi

1

|z−ζ|, whereτiis eitherτi(z,ε)orτi(ζ,ε).

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An elementary calculation shows that:

Lemma 3.9. For z∈Ω,δ small and0≤µ<1,

(3.5) ^Z

P(z,δ)

dλ(ζ)

|z−ζ|^1+µ .τn(z,δ)^1−µ

n−1

∏j=1

τ²_j(z,δ),

and, forα>0,

(3.6) ^Z

P(ζ,δ)

δ_Ω^α−1(z)

|z−ζ| dλ(z).δ^α−1 α τn(ζ,δ)

n−1

∏j=1

τ²_j(ζ,δ).

3.1. Proof of Theorem 2.1.

Proof of (1) of Theorem 2.1. It is based on a version of a classical operator estimate which can be found, for example, in Appendix B of the book of M. Range [Ran86]:

Lemma 3.10. LetΩbe a smoothly bounded domain inCⁿ. Letµandνbe two positive measures onΩ. Let K be a measurable function onΩ×Ω. Assume that there exists a positive numberε0>0, a positive constant C and a real number s≥1such that:

(1) ^R_Ω|K(z,ζ)|^sδ_Ω^−ε(ζ)dµ(ζ)≤Cδ_Ω^−ε(z), (2) ^R_Ω|K(z,ζ)|^sδ_Ω^−ε(z)dν(z)≤Cδ_Ω^−ε(ζ),

for allε≤ε₀, whereδ_Ωdenotes the distance to the boundary ofΩ. Then the linear operator T defined by

T f(z) = Z

ΩK(z,ζ)f(ζ)dµ(ζ)

is bounded from L^p(Ω,µ)to L^q(Ω,ν)for all1≤p,q<∞such that¹_q=¹_p+¹_s−1.

Short proof. This is exactly the proof given by M. Range in his book: letεbe sufficiently small. Writing

K f =

K^sf^pδ^ε

p−1

p q

Ω (ζ) ¹/q

K¹⁻^q^sδ^−ε

p−1 p

Ω

f¹⁻

p q,

Hölder’s inequality (with ¹_q+^p−1_p +^s−1_s =1) gives

|T f(z)| ≤ _Z

Ω|K(z,ζ)|^sδ^ε

p−1

p q

Ω (ζ)|f|^p(ζ)dµ(ζ) ¹/q

_Z

Ω|K(z,ζ)|^sδ_Ω^−ε(ζ)

^p−1_p _Z

Ω|f(ζ)|^pdµ(ζ) ^s−1_s

. The first hypothesis of the lemma gives (forε≤ε0)

|T f(z)|^q≤C _Z

Ω|K(z,ζ)|^sδ^ε

p−1

p q

Ω (ζ)δ^−ε

p−1

p q

Ω (z)|f|^p(ζ)dµ(ζ)

Z

Ω|f(ζ)|^pdµ(ζ) q^s−1_s

. Integration with respect to the measuredν(z)gives (using the second hypothesis of the lemma withε^p−1_p q≤ε0)

Z

Ω|T f(z)|^qdν(z)≤C² Z

Ω|f|^pdµ ^q/p

.

Applying this lemma to the operatorT_K(formula (3.1)) withµ=δ_Ω^γdλ andν=δ_Ω^γ^′dλ, the required estimates onK_N¹are summarized in the following Lemma: