If Ω is smooth, we shall give sufficient conditions for compactness of the∂-Neumann operator

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

COMPACTNESS OF THE CANONICAL SOLUTION OPERATOR ON LIPSCHITZ q-PSEUDOCONVEX BOUNDARIES

SAYED SABER

Communicated by Jerome A. Goldstein

Abstract. Let Ω⊂Cⁿbe a bounded Lipschitzq-pseudoconvex domain that admit good weight functions. We shall prove that the canonical solution op- erator for the∂-equation is compact on the boundary of Ω and is bounded in the Sobolev spaceW_r,s^k (Ω) for some values ofk. Moreover, we show that the Bergman projection and the∂-Neumann operator are bounded in the Sobolev spaceW_r,s^k (Ω) for some values ofk. If Ω is smooth, we shall give sufficient conditions for compactness of the∂-Neumann operator.

1. Introduction

Pseudoconvex domains are central objects in several complex variables analysis as they are natural domains for existence of holomorphic functions. It turns out that boundaries of domains play a leading role in the theory of several complex variables.

In this article, we discuss the existence of a compact canonical solution operator

∂^∗N to the∂-equation on the boundary of a Lipschitzq-pseudoconvex domain that admits a good weight function. The connection between finite type and good weight functions was first observed by Catlin [8, 9]. Straube [41] showed that Catlin’s result could be used to construct useful weight functions on certain Lipschitz domains.

Harrington-Zeytuncu [26] showed that on bounded Lipschitz pseudoconvex domains that admit good weight functions, the∂-Neumann operatorsN, ∂N and ∂^∗N are bounded onL^p spaces, for some values ofpgreater than 2. Shaw [40] constructed a solution to the tangential Cauchy-Riemann operator∂_b that is regular onL² on Lipschitz domains with plurisubharmonic defining functions. In [39], the author extended this result to Lipschitzq-pseudoconvex domains. The first main result in this article proves the compactness of this solution.

Theorem 1.1. LetΩ⊂Cⁿ be a bounded Lipschitzq-pseudoconvex domain and let 1≤q≤n. Letρbe a defining function ofΩsatisfying

i∂∂ρ≥i(−ρ)φ(−ρ)∂∂|z|² onΩ, for some positive function φ∈C(0,∞)satisfying

lim

x→0⁺φ(x) = +∞.

2010Mathematics Subject Classification. 35J20, 35J25, 35J60.

Key words and phrases. Lipschitz domain;q-pseudoconvex domain;∂;∂b;∂-Neumann.

c

2019 Texas State University.

Submitted May 8, 2018. Published April 10, 2019.

1

Thus, there exists a compact solution operatorS:L²_r,s(bΩ)∩ker(∂b)→L²_r,s−1(bΩ) such that ∂_bS=I, for everys≥q.

When Ω hasC¹-boundary and has a plurisubharmonic defining function on the boundarybΩ of Ω, Boas-Straube [5] proved that the Bergman projection maps the Sobolev space W^k(Ω) into itself for any k > 0. On C²-pseudoconvex domains, Diederich-Fornaess [15] constructed a global defining functionρso that−(−ρ)^α is a bounded plurisubharmonic function for some 0< α <1. Berndtsson-Charpentier [3] showed that in such cases the Bergman projection and the canonical solution operator ∂^∗N are regular in any Sobolev space W^k(Ω), for 0≤k < α/2 (see also [7]). Harrington [25] showed that the result of Diederich-Fornaess and Berndtsson- Charpentier still holds when the boundary is only Lipschitz. However, Diederich- Fornaess [16] used worm domain to show that for any 0 < α <1, one can find a smooth pseudoconvex domain where−(−ρ)^αis not plurisubharmonic for any global defining function ρ. Barrett [2] showed that the Bergman projection on a smooth worm domain does not mapW^k into W^k for some values ofk. OnC²-weakly q- convex domains, Herbig-McNeal [28] constructed a global defining functionρso that

−(−ρ)^α is a bounded strictly plurisubharmonic function for some 0< α < 1. In [35], the author showed that in such cases the Bergman projection and the canonical solution operator ∂^∗N are regular in any Sobolev space W^k(Ω), for 0≤k < α/2.

The second main result in this article extends the result of Berndtsson-Charpentier to all Lipschitzq-pseudoconvex domains.

Theorem 1.2. LetΩ⊂Cⁿ be a bounded Lipschitzq-pseudoconvex domain and let 1≤q≤n. Suppose that there exists a Lipschitz defining functionρforΩsuch that there exists some0< α <1 with

i∂∂(−(−ρ)^α)≥0 onΩ. (1.1)

Thus, for 0 < k < α/2 and for q+ 1 ≤ s ≤n−1, the Bergman projection and the canonical solution operator for the ∂-equation are bounded in the Sobolev space W_r,s^k (Ω).

Cao-Shaw-Wang [7] extend Berndtsson-Charpentier’s result to obtain estimates for the∂-Neumann operator. In [36] the author proved this result in the case of logδ-pseudoconvexity in a K¨ahler manifold for forms with values in a holomorphic vector bundle.

Theorem 1.3. LetΩ⊂Cⁿ be a bounded Lipschitzq-pseudoconvex domain and let 1 ≤ q ≤ n. Suppose that there exists a Lipschitz defining function ρ for Ω such that there exists some 0 < α < 1 satisfies (1.1). Thus, for 0 < k < α/2 and for q+ 1≤s≤n−1, the∂-Neumann operator is bounded in the Sobolev spaceW_r,s^k (Ω).

Also, we provide sufficient conditions for compactness of the∂-Neumann prob- lem. Our motivation for studying compactness of the∂-Neumann problem comes from its connections to the geometry of the boundaries ofq-pseudoconvex domains.

There have been two different approaches for compactness of the∂-Neumann prob- lem. The first is a potential theory approach. Catlin [8] introduced Property (P) and showed that it implies the compactness of the ∂-Neumann problem. McNeal [32] introduced Property ( ˜P) and showed that it still implies compactness of the

∂-Neumann problem. The second approach is geometric in nature. Straube [42]

introduced a geometric condition that implies compactness of the ∂-Neumann op- erator on domains inC². This problem was considered in [18, 19, 20, 32, 24]. Some recent work on compactness of the ∂-Neumann operator, for non-pseudoconvex domains, can be found in [37, 38].

Theorem 1.4. Let Ω be a smooth boundedq-pseudoconvex domain inCⁿ and let 1≤q≤n. IfΩsatisfies a McNeal’s Property( ˜P), thenN is compact (in particular, continuous) as an operator fromW_r,s^k (Ω) to itself, for allk≥0 and fors≥q.

2. Preliminaries

Let (z1, . . . , zn) be the complex coordinates forCⁿ. Let Ω⊂Cⁿ be a bounded domain with C² boundary andρbe its C² defining function. For 0≤r, s≤n, an (r, s)-formuon Ω, can be expressed as

u=

0

X

I,J

uI,Jdz^I∧dz^J,

where I = (i1, . . . , ir) and J = (j1, . . . , js) are multi-indices and dz^I = dzi₁ ∧

· · · ∧dz_ir, d¯z^J =d¯z_j1 ∧ · · · ∧d¯z_js. The notation P0

means the summation over strictly increasing multi-indices. Denote by C^∞(Cⁿ) the space of complex-valued C^∞ functions on Cⁿ and C_r,s^∞(Cⁿ) the space of complex-valued differential (r, s)- forms of class C^∞ on Cⁿ. Let C_r,s^∞(Ω) = {u

_Ω : u ∈ C_r,s^∞(Cⁿ)} be the subspace ofC_r,s^∞(Ω) whose elements can be extended smoothly up to the boundarybΩ. Let D(Cⁿ) be the space of C^∞-functions with compact support in Cⁿ. A form u ∈ C_r,s^∞(Cⁿ) is said to be has compact support in Cⁿ if its coefficients belongs to D(Cⁿ). The subspace ofC_r,s^∞(Cⁿ) which has compact support inCⁿ is denoted by Dr,s(Cⁿ). For u, v∈C_r,s^∞(Cⁿ), the local inner product (u, v) is denoted by

(u, v) =

0

X

I,J

uI,JvI,J.

Letφ:Cⁿ→R⁺ be a plurisubharmonicC²-weight function and define the space L²(Ω, φ) ={u: Ω→C:

Z

Ω

|u|²e^−φdV <∞},

wheredV denotes the Lebesgue measure. Denote the inner product and the norm inL²(Ω, φ) by

hu, viφ= Z

Ω

uve^−φdV and kukφ= Z

Ω

|u|²e^−φdV.

We also have the inner product and norm defined on the boundary:

hu, vibφ=hu, viL²(bΩ,φ)= Z

bΩ

uve^−φdS, kukbφ=kukL²(bΩ,φ)=

Z

bΩ

|u|²e^−φdS.

We will typically abbreviate hu, vi0 as hu, vi. Recall that L²_r,s(Ω, φ) the space of (r, s)-forms with coefficients inL²(Ω, φ). If u, v∈L²_r,s(Ω, φ), theL²-inner product

and norms are defined by hu, viφ,Ω=

Z

Ω

(u, v)e^−φdV = Z

Ω

tu∧?ve^−φ and kuk²_φ,Ω=hu, uiφ,Ω, where?:C_r,s^∞(Cⁿ)→C_n−s,n−r^∞ (Cⁿ) is the Hodge star operator such that? u=?u (that is? is a real operator) and? ? u= (−1)^r+su. Set

Q(u, u) =kuk²+k∂uk²+k∂^∗uk².

For a formu, the vector of allm-th derivatives of all coefficients ofuwill be denoted

∇^mu(we treat∇⁰as the identity). Ifρis the distance function forbΩ, for any real number−1≤k≤1 and integerm >0, one defines

hu, vi_W(k)(Ω)= Z

Ω

(u, v)(ρ(z))^−2kdV, kuk²_W(k)(Ω)=hu, ui_W(k)(Ω), hu, vi_W(m,k)(Ω)=

(h∇^mu,∇^mvi_W(k)(Ω)+h∇^m−1u,∇^m−1vi+hu, vi whenk≤0, h∇^mu,∇^mvi_W(k)(Ω)+hu, vi whenk >0,

kuk²_W(m,k)(Ω)=hu, ui_W(m,k)(Ω). The corresponding function spaces are defined by

W_r,s^(k)(Ω) ={u∈L²_r,s(Ω) :kuk²_W(k)(Ω)<∞}, W_r,s^(m,k)(Ω) =

(u∈W_r,s^m−1(Ω) :kuk²_W(m,k)(Ω)<∞ whenk≤0, {u∈W_r,s^m(Ω) :kuk²_W(m,k)(Ω)<∞} whenk >0.

Leta= (a1, . . . , an) be a multi-index; that is, a1, . . . , an are nonnegative integers.

Forx∈Rⁿ, one definesx^a=x^a₁¹. . . x^a_nⁿ andD^a is the operator D^a =1

i

∂

∂x1

^a1

. . . 1 i

∂

∂xn

^an

.

Denote bySthe Schwartz space of rapidly decreasing smooth functions onRⁿ; that is,Sconsists of all functionsuwhich are smooth onRⁿ with sup_x∈Rn|x^aD^bu(x)|<

∞for all multi-indicesa,b. The Fourier transform ˆuof a functionu∈ S is defined by

ˆ

u(ξ) = (2π)^−n/2 Z

Rⁿ

u(x)e^−ix·ξdx, where x·ξ = Pn

j=1xjξj and dx = dx1 ∧ · · · ∧dxn with x = (x1, . . . , xn) and ξ= (ξ1, . . . , ξn). Ifu∈ S, then ˆu∈ S. The Sobolev space W^k(Rⁿ),k∈R, is the completion ofS under the Sobolev norm

kuk²_Wk(Rⁿ)= Z

Rⁿ

(1 +|ξ|²)^k|ˆu|²dξ.

Denote byW^k(Ω),k≥0, the space of the restriction of all functionsu∈W^k(Cⁿ) = W^k(R²ⁿ) to Ω and

kuk_Wk(Ω)= inf{kfk_Wk(Cⁿ), f ∈W^k(Cⁿ), f|Ω=u}

the W^k(Ω)-norm. Denote by W₀^k(Ω) the completion of D(Ω) under the W^k(Ω)- norm andW_r,s^k (Ω),k∈R, the Hilbert spaces of (r, s)-forms withW^k(Ω)-coefficients

and their norms are denoted by kuk_Wk(Ω). In addition, for any (1,1)-form Θ = Θ_{i j}dzⁱ∧dz^j we have

(u, v)^∗_Θ=u_iIΘ_{i j}v_iI.

The ∗ is used to emphasize that these norms are dual to the norms defined by Demailly in [13].

Let ∂ : L²_r,s(Ω) → L²_r,s+1(Ω) be the maximal closed extensions of the Cauchy- Riemann operator∂:C_r,s^∞(Ω)→C_r,s+1^∞ (Ω) and let∂^∗ be its Hilbert space adjoint.

Define

H²(Ω) ={u∈L²(Ω) :4u= 0 on Ω}, H^r,s(Ω) ={u∈L²_r,s(Ω) :∂u=∂^∗u= 0 on Ω},

where 4is the real Laplacian operator. The ∂-Neumann operator N :L²_r,s(Ω)→ L²_r,s(Ω) is defined as the inverse of the restriction of the complex Laplacian =

∂∂^∗+∂^∗∂ to (H^r,s(Ω))^⊥. Note that N may not always exist. The Bergman pro- jectionBis the orthogonal projection from the space of square integrable functions onto the space of square integrable holomorphic functions on a domain. For any 0≤r≤n and 1≤s≤n, denote by B :L²_r,s(Ω)→ker∂ the Bergman projection operator.

Definition 2.1 ([8]). A domain Ω has Property (P), if for every positive number M there exists a smooth plurisubharmonic functionλon Ω such that 0≤λ≤1 on Ω andi∂∂λ≥iM ∂∂|z|² on the boundarybΩ.

McNeal [32] defined Property ( ˜P) (a generalization of Catlin’s Property (P)) as follows:

Definition 2.2. A domain Ω has the McNeal Property ( ˜P) if for every positive numberM there existsλ=λM ∈C²(Ω) such that

(1) |∂λ|_i∂∂λ≤1;

(2) the sum of anyqeigenvalues of the matrix _∂z^∂2λ

k∂z_k

(z)≥M, for allz∈bΩ.

A bounded domain is called Lipschitz if locally the boundary of the domain is the graph of a Lipschitz function. The defining function associated with a Lipschitz domain is called a Lipschitz defining function.

Definition 2.3. A bounded Lipschitz domain Ω inCⁿ is said to have a Lipschitz defining function if there exists a Lipschitz function ρ:Cⁿ →Rsatisfies ρ <0 in Ω,ρ >0 outside Ω and

C₁<|dρ|< C₂ a.e. onbΩ, whereC₁, C₂ are positive constants.

Lemma 2.4([23]). LetΩ⊂Cⁿbe a bounded Lipschitz domain. For any0< k < ¹₂, one obtains W^k(Ω)⊂W^(k)(Ω).

Lemma 2.5([30]). LetΩ⊂Cⁿ be a bounded Lipschitz domain. For some constant 0≤k≤1and integer m≥0, one obtains

H²(Ω)∩W^m+k(Ω) =H²(Ω)∩W^(m+1,k−1)(Ω).

Definition 2.6. Let Ω be an open domain. A function ϕ: Ω → R is called an exhaustion function for Ω if the closure of{x∈Ω|ϕ(x)< c}is compact for all real c.

Now, we recall the following definition of q-subharmonic functions which has been introduced by Ahn-Dieu [1] (also see [29]).

Definition 2.7. Let Ω be a bounded domain inCⁿ and let qbe an integer with 1 ≤q ≤ n. A semicontinuous function η defined in Ω is called a q-subharmonic function if for every q-dimension spaceLin Cⁿ, η|L is a subharmonic function on L∩Ω. This means that for every compact subsetK⊂L∩Ω and every continuous harmonic functionhonKsuch thatη≤honbK, thenη≤honK.

The functionη is called strictlyq-subharmonic if for everyU ⊂Ω there exists a constantCU >0 such thatη−CU|z|² isq-subharmonic.

Proposition 2.8 ([1]). Let Ω be a bounded domain inCⁿ and let q be an integer with1≤q≤n. Letη: Ω→[−∞,∞)be aC²smooth function. Thus, the following statements are equivalent:

(1) η is aq-subharmonic function.

(2) For every smooth(r, s)-formf =P

I,JfI,Jdz^I∧dz^J, and for s≥q,

0

X

I,K n

X

j,k=1

∂²η

∂z^j∂z^kfI,jKf_I,kK ≥0. (2.1) Definition 2.9. A Lipschitz domain Ω⊂Cⁿis said to be (strictly)q-pseudoconvex if there is a (strictly)q-subharmonic exhaustion Lipschitz function on Ω.

Definition 2.10.

(i) A C² smooth function u on U ⊂ Cⁿ is called q-plurisubharmonic if its complex Hessian has at least (n−q) non-negative eigenvalues at each point ofU.

(ii) Ann-subharmonic function is just subharmonic function in usual sense. An upper semicontinuous function onU is plurisubharmonic exactly when it is 1-subharmonic.

Example 2.11([22]). Let Ω⊂Cⁿbe a bounded domain satisfy theZ(q) condition, that is, the Levi form of a smooth defining function of Ω has, at every boundary point of Ω, at leastn−qpositive or at leastq+ 1 negative eigenvalues. Thus Ω is strictlyq-pseudoconvex.

Remark 2.12. A domain Ω⊂ Cⁿ is pseudoconvex if and only if it is 1-pseudo- convex, since 1-subharmonic function is just plurisubharmonic.

Remark 2.13 ([22]). If Ω⊂Cⁿ is aq-pseudoconvex domain, 1≤q≤n, then the following hold

(1) IfbΩ is of classC², thus by (2.1), Ω is weaklyq-convex;

(2) if q≤q⁰, Thusq-pseudoconvexity implies q⁰-pseudoconvexity.

Proposition 2.14([22]). LetΩ be a domain inCⁿ and let1≤q≤n. Thus, one obtains:

(i) If {ηj}^∞_j=1 is a decreasing sequence ofq-subharmonic functions. Thusη= lim_j→+∞η_j is aq-subharmonic function;

(ii) letχbe a nonnegative smooth function inCⁿ vanishing outside the unit ball and satisfyingR

CⁿχdV = 1. Iff is aq-subharmonic function, one defines f(z) = (f∗χ)(z) =

Z

B(0, )

f(z−w)χ(w)dVw, ∀z∈Ω,

where χ(z) = χ(z/)/||²ⁿ and Ω = {z ∈ Ω :d(z, bΩ) > }. Thus f is smooth q-subharmonic onΩ, andf↓f as↓0;

(iii) if η ∈ C²(Ω) such that _∂z^∂j²∂z^ηk(z) = 0 for all j 6= k and z ∈ Ω. Thus η is q-subharmonic if and only if P

j,k∈J

∂²η

∂z^j∂z^k(z)≥0, for all |J| =s, for s≥qand for all z∈Ω.

If Ω is a bounded Lipschitz domain with distance function ρ. We equip the boundary bΩ with the induced metric fromCⁿ. Let C^∞(bΩ) be the space of the restriction of all smooth functions in Cⁿ to bΩ. L²(bΩ) denote the space of L² functions on the boundary of Ω, and ˜L²_r,s(bΩ) denote the space of (r, s)-forms in Ω such that the restrictions of the coefficients to bΩ are in L²(bΩ). Fix p ∈ bΩ.

Thus for some neighborhoodU ofplocally choose an orthonormal coordinate patch {dz1, . . . , dz_n} defined almost everywhere inU∩Ω such that dz_n=−∂ρa.e. Note that |∂ρ|= ¹₂ because we are using the metric where|dzj| = 1, which is half the size induced by the usual Euclidean metric on Rⁿ. Define L²_r,s(bΩ)⊂L˜²_r,s(bΩ) as the space of allf ∈L˜²_r,s(bΩ) such thatdz_n∨f = 0 almost everywhere onbΩ.

Definition 2.15. Foru∈L²_r,s(bΩ) andf ∈L²_r,s+1(bΩ),uis in dom∂_band∂_bu=f if

Z

bΩ

u∧∂φ dS= (−1)^r+s Z

bΩ

f∧φ dS, for everyφ∈C_{n−r,n−s−1}^∞ (Cⁿ).

Thusuis said to be in dom∂b and∂bu=f.

Since∂²= 0, it follows that∂²_b = 0. Thus∂b is a complex and one obtains 0→L²_r,0(bΩ)→^∂b L²_r,1(bΩ)→^∂b L²_r,2(bΩ)→^∂b . . .→^∂b L²_r,n−1(bΩ)→0.

The ∂_b operator is a closed, densely defined, linear operator from L²_r,s−1(bΩ) to L²_r,s(bΩ), where 0≤r≤n, 1≤s≤n−1.

Definition 2.16. dom∂^∗_b is the subset of L²_r,s(bΩ) composed of all forms f for which there exists a constantC >0 satisfies

|hf, ∂buiL²(bΩ)| ≤CkukL²(bΩ), for allu∈dom∂_b.

For allf ∈dom∂^∗_b, let∂^∗_bf be the unique form in L²_r,s(bΩ) satisfying h∂^∗_bf, uiL²(bΩ) =hf, ∂buiL²(bΩ),

for all u ∈ dom∂b. The ∂b Laplacian operator ^b = ∂b∂^∗_b +∂^∗_b∂b : dom^b → L²_r,s(bΩ) is defined on domb = {u ∈ L²_r,s(bΩ) : u ∈ dom∂_b ∩dom∂^∗_b : ∂_bu ∈ dom∂^∗_b and∂^∗_bu∈dom∂_b}. The∂_bLaplacian operator is a closed, densely defined self-adjoint operator. The space of harmonic formsH^r,s_b (bΩ) is denoted by

H^r,s_b (bΩ) ={u∈domb:∂_bu=∂^∗_bu= 0}.

The space H^r,s_b (bΩ) is a closed subspace of dom^b since^b is a closed operator.

The∂b-Neumann operatorNb:L²_r,s(bΩ)→L²_r,s(bΩ) is defined as the inverse of the restriction of^b to (H^r,s_b (bΩ))^⊥.

The Bochner-Martinelli-Koppelman kernel on Lipschitz domains is defined in [27] for (r, s)-forms as follows. Define

(ζ−z, dζ) =

n

X

j=1

(ζ_j−z_i)dζ_j,

(dζ−dz, dζ) =

n

X

j=1

(dζ_j−dz_j)dζ_j,

where (ζ−z) = (ζ1−z1, . . . , ζn −zn), dζ = (dζ1, . . . , dζn). Thus, the Bochner- Martinelli-Koppelman kernelK(ζ, z) is defined by

K(ζ, z) = 1 (2πi)ⁿ

(ζ−z, dζ)

|ζ−z|² ∧(dζ−dz, dζ)

|ζ−z|²

n−1

=

n−1

X

s=0

Ks(ζ, z),

whereKs(ζ, z)is the the component ofK(ζ, z); that is, an (r, s) inzand of degree (n−r, n−s) inζ. Whenn= 1,K(ζ, z) = (2πi)⁻¹dζ/(ζ−z) is the Cauchy kernel.

As in the Cauchy integral case, for any f ∈L²_r,s(bΩ) the Cauchy principal value integralKbf is defined as

Kbf(z) = lim

→0⁺

Z

bΩ

|ζ−z|>

Ks(ζ, z)∧f(ζ),

whenever the limit exists. Denote byν_zthe outward unit normal tobΩ atz. Since bΩ is Lipschitz,νzexists almost everywhere onbΩ. Thus, forz∈bΩ, one defines

K_b⁻f(z) = lim

→0⁺

Z

bΩ

Ks(·, z−υz)∧f, K_b⁺f(z) = lim

→0⁺

Z

bΩ

K_s(·, z+υ_z)∧f.

The properties of the Bochner-Martinelli-Koppelman kernel and the related trans- forms are developed on smooth domains in [11], and on Lipschitz domains in [40].

In [23, Lemma 4.1.1] we find the following result.

Lemma 2.17. LetΩbe a bounded domain in Cⁿ. Thus, for anyf ∈L²_r,s(bΩ), one obtains

K_b⁻f = 1

2f+K_bf, K_b⁺f =−1

2f+K_bf, f =K_b⁻f−K_b⁺f,

(2.2)

almost everywhere on bΩand

kKbfk².kfk².

3. A priori estimates for the the ∂-Neumann operator In this section, we find a priori estimates that we need in the later sections.

Lemma 3.1([43]). LetΩ⊂Cⁿ be a bounded domain withC² boundary andρbe a C²defining function ofΩ. Letσbe a real-valued function that is twice continuously differentiable onΩ, withσ≥0. Then, forf ∈C_r,s^∞(Ω)∩dom∂^?_φ with1≤s≤n−1, one obtains

k√

σ ∂fk²_φ+k√

σ ∂^∗_φfk²_φ

=X

I,K n

X

j, k=1

Z

bΩ

σ ∂²ρ

∂z^j∂z^kfI,jKf_I,kKe^−φdS

+X

I,J n

X

k=1

Z

Ω

σ

∂fI,J

∂z^k

2e^−φdV

+ 2 ReD X

I,K n

X

j=1

∂σ

∂z^j f_I,jKdz^I∧dz^K, ∂^∗_φfE

φ

+X

I,K n

X

j, k=1

Z

Ω

σ ∂²φ

∂z^j∂z^k − ∂²σ

∂z^j∂z^k

f_I,jKf_I,kKe^−φdV.

(3.1)

The case σ≡1andφ≡0 is the classical Kohn-Morrey formula.

Proposition 3.2([39]).LetΩ⊂Cⁿbe aq-pseudoconvex domain and let1≤q≤n.

Thus, for any s≥q, there exists a bounded linear operatorN :L²_r,s(Ω)→L²_r,s(Ω) satisfies the following properties:

(i) rangeN ⊂dom,N=I ondom;

(ii) for anyf ∈L²_r,s(Ω), one obtains f =∂ ∂^∗N f⊕∂^∗∂N f; (iii) ∂N =N ∂ ondom∂,q≤s≤n−1,n≥2;

(iv) ∂^∗N=N ∂^∗ ondom∂^∗,q+ 1≤s≤n;

(v) N,∂N and∂^∗N are bounded operators with respect to theL²-norms. That is

kN fk ≤e d² s

kfk,

k∂N fk+k∂^∗N fk ≤2 re d²

s kfk; (vi) the Bergmann projectionB is given by

B=Id−∂^∗N ∂.

Corollary 3.3. For every f ∈ L²_r,s(Ω)∩ker∂ and for s ≥ q. Thus u = ∂^∗N f satisfying∂u=f in the distribution sense inbΩwith

kuk ≤Ckfk,

where C depends only on the Lipschitz constant and the diameter of Ω, but is independent of f. uis the unique solution to ∂u =f that is orthogonal to ker∂, u=∂^∗N f =Sf is called the canonical solution operator for the ∂-equation.

Lemma 3.4 ([23]). Let φ∈C(0,∞)such that φ(x)>0 for allx >0and lim

x→0⁺φ(x) =∞.

Thus there existsφ˜∈C¹(0,∞) such that

(i) inf_(0,∞)φ(x)≤φ(x)˜ < φ(x) for allx >0, (ii) lim_x→0+φ(x) = +∞,˜

(iii) lim_x→0+φ˜⁰(x) =−∞, (iv) lim_x→0+xφ˜⁰(x) = 0.

Lemma 3.5([33, Lemma 1.1]). Let Ωbe a bounded Lipschitz domain inRⁿ. Thus Ω has a Lipschitz defining function ρ. Furthermore, the distance function to the boundary is comparable to|dρ|for any Lipschitz defining functionρnear the bound- ary.

Proposition 3.6 ([23, Prop. 3]). Let Ω ⊂ Cⁿ be a C²-domain with a defining function ρsuch that |dρ|bΩ = 1and a weight function ϕ such that e^−ϕ ∈ C²(Ω).

Thus for anyg∈C_r,s² (Ω),1≤s≤n, one obtains k∂gk²_ϕ+h∂ϕ∨g, ∂ρ∨gi_bϕ

=kϑgk²_ϕ−2 Reh∂ϑg, giϕ+k∇gk²_ϕ+kgk^∗2_∂∂ϕ,ϕ− k∂ϕ∨gk²_ϕ+kgk^∗2_b∂∂ρ,ϕ +h∂ρ∨g, ∂^∗gibϕ− h∂(∂ρ∨g, gibϕ

(3.2)

Lemma 3.7 ([39]). Let Ω⊂ Cⁿ be a bounded Lipschitz q-pseudoconvex domain.

There exists an exhaustion{Ων} of Ωsuch that

(i) there exists a Lipschitz function ρ:Cⁿ →Rsuch that ρ <0 in Ω, ρ >0 outsideΩand satisfiesC1<|dρ|< C2 a.e. onbΩ;

(ii) {Ων} is an increasing sequence of relatively compact subsets ofΩandΩ =

∪νΩν;

(iii) eachΩν,ν= 1,2, . . ., is strictlyq-pseudoconvex domains, i.e., eachΩν has aC^∞strictlyq-subharmonic defining functionρν on a neighbourhood of Ω, such that

0

X

I,K

X

j, k

∂²ρν

∂z^j∂z^kfI,jKf_I,kK ≥C0|f|²,

forf ∈C_r,s^∞(Ων)∩ dom∂^∗_ν withs≥q andC0>0 is independent ofν; (iv) there exist positive constants C₁, C₂ such that C₁ ≤ |∇η_ν| ≤ C₂ on bΩ_ν,

whereC₁,C₂ are independent of ν.

The proof of the following proposition follows the ideas in Bonami-Charpentier [6] (see also [23, Theorem 3.5.1]).

Proposition 3.8. LetΩ⊂Cⁿ be a bounded Lipschitzq-pseudoconvex domain and let1≤q≤n. Letρbe a defining function ofΩsatisfying

i∂∂ρ≥i(−ρ)φ(−ρ)∂∂|z|² onΩ, for some positive function φ∈C(0,∞)satisfying

lim

x→0⁺φ(x) = +∞.

Thus, for q+ 1 ≤ s ≤ n−1 and for all f ∈ Wr,s^1/2(Ω)∩(ker∂)^⊥ such that k∂fk²_W1/2(Ω)<∞, one obtains

k∂^∗N fk²_W1/2(Ω).εkfk²_W1/2(Ω)+Cεkfk²_W−1(Ω). (3.3)

Proof. Let Ω be a strictlyq-pseudoconvex domain with smooth boundary. Letδbe the distance function of Ω. As in [10, Lemma 4.3] (see also [6]), a special extension operator on bΩ is constructed as follows. Let f ∈ Wr,s^1/2(bΩ) be any form on bΩ withq+ 1≤s≤n−1, and let ˜f ∈W_r,s¹ (Ω) be any extension off to the interior of Ω (i.e.f is the boundary trace of ˜f). One can defineT:Wr,s^1/2(bΩ)→L²_r,s+1(Ω) by

T f =−2∂[ϑ, N](∂δ∧f˜).

This definition does not depend on the choice of ˜f, since when f ≡ 0, we have

∂δ∧f˜∈dom∂^∗ and hence [ϑ, N](∂δ∧f˜) = 0. Clearly∂T f = 0, and

∂ϑT f=−2(∂ϑN−∂N ϑ)(∂δ∧f˜) = 0.

Together, these imply thatT f = 0, soT f must have harmonic coefficients. Using the boundary conditions for dom= rangeN, one can also see that

−∂δ∨T f|bΩ= 2∂δ∨N(∂δ∧f˜)|bΩ= 2∂δ∨∂δ∧f

so the boundary value of −∂δ∨T f is identical to the tangential component of f. The adjoint T^∗ : L²_r,s+1(Ω) → Wr,s^−1/2(bΩ) is precisely the restriction of ∂^∗N to the boundary of Ω. The adjoint of the trace of the Bergman projection B is precisely −ϑT on functions, while on forms −ϑT will be the adjoint of the trace of 2∂δ∨∂δ∧Bf. The properties ofT immediately give us−ϑT f ∈ker∂∩kerϑ.

Assume that ∂ρ = −|dρ|∂δ. Then, for g ∈ L²_r,s(Ω) and by applying (3.2) with ρ/|dρ|as our defining function andϕ=−log(−ρ) to obtain

k∂gk²_W(−1/2)(Ω)+k |dρ|^−1/2∂ρ∨gk²_L2(bΩ)

≥ kϑgk²_W(−1/2)(Ω)−2 Reh∂ϑg, gi_W(−1/2)(Ω)+kp

ϕ(−ρ)gk²_W(−1/2)(Ω).

(3.4) Applying tog=T f gives us

k |dρ|^−1/2∂ρ∧fk²_L2(bΩ)≥ kϑT fk²_W(−1/2)(Ω)+kp

ϕ(−ρ)T fk²_W(−1/2)(Ω). (3.5) To prove (3.3), we approximate Ω as Lemma 3.7 by a sequence of subdomains Ων ={ρ < −ν} such that each Ων is strictlyq-pseudoconvex domains with C^∞ smooth boundary, i.e., each Ων has aC^∞ strictlyq-subharmonic defining function ρν such that (ii) and (iii) in Lemma 3.4. Thus, we can apply (3.4) and (3.5) on each Ων. We useTν, T_ν^∗ and Nν, to denote the corresponding operators on each Ων. Then, from (3.5), one obtains

k |dρν|^−1/2∂ρν∧fk²_L2(bΩ)≥ kϑνTνfk²_W(−1/2)(Ων)+kp

ϕ(−ρν)Tνfk²_W(−1/2)(Ων). (3.6) Passing to the limit, one obtains from (3.6) that

k |dρ|^−1/2∂ρ∧fk²_L2(bΩ)≥ kϑT fk²_W(−1/2)(Ω)+kp

ϕ(−ρ)T fk²_W(−1/2)(Ω). (3.7) Using that for harmonic functionh,

khk²_W−1/2(Ω)&khk²_W(−1/2)(Ω), for a proof see [10, Lemma 2.2], or [11]. Givenε >0, set

U_ε:={z∈Ω :ϕ(−ρ)> ε⁻¹}.

SinceT f has harmonic coefficients, we may use estimate (3.7) and interior regularity for harmonic functions to obtain

k∂ρ∧fk²_L2(bΩ)≥ε⁻¹kT f|U_εk²_W−1/2(Ω)+C_ε⁻¹kT f|_U\Uεk²_W1(Ω). By duality, one obtains

εkfk²_W1/2(Ω)+C_εkfk²_W−1(Ω)&k∂^∗N fk²_L2(bΩ). A result of Dahlberg (see [12]) tells us that for harmonic functionh,

khk²

W^(1,−12)(Ω)&khk²_L2(bΩ)&k∇hk²_W(−1/2)(Ω). Combining this with Lemma 2.4, one can show that

εkfk²_W1/2(Ω)+Cεkfk²_W−1(Ω)&k∂^∗N fk²_W1/2(Ω).

4. Proof of Theorem 1.1

In this section, we use the estimates in Section 3 to construct a compact solution operator to the ∂b operator. When the domain satisfies the additional conditions of Proposition 3.8, one can use the new jump formula forK(ζ, z), to show that we have a compact solution operator.

Letf ∈L²_r,s(bΩ)∩ker∂b. Choose a ballD so that Ω⊂D. Set Ω⁺=D\Ω. By [11, Lemma 9.3.5], (see also [40, Lemma 4.1]), there exist∂-closed forms

f⁺(z) =K⁺f(z), f⁺(z)∈C_r,s¹ (Ω⁺)⊂W_r,s¹ (Ω⁺), f⁻(z) =K⁻f(z), f⁻(z)∈C_r,s¹ (Ω)⊂W_r,s¹ (Ω),

such thatf =f⁻−f⁺ onbΩ (in the sense of traces of the coefficients, but also in the sense of restrictions of forms: i.e. the normal components off⁺ andf⁻ cancel each other out at points ofbΩ). Moreover,

kf⁺k_W1/2(Ω⁺)≤Ckfk_L2(bΩ), kf⁻k_W1/2(Ω)≤CkfkL²(bΩ).

Furthermore,f⁻andf⁺have harmonic coefficients with boundary values inL²(bΩ), so they are both inW^1/2.

On Ω, one can setu⁻ =∂^∗N f⁻, and for anyε >0 we haveCε>0 such that ku⁻k²_W1/2(Ω)≤εkf⁻k²_W1/2(Ω)+C_εkf⁻k²_W−1(Ω)

≤εkfk²_L2(bΩ)+Cεkf⁻k²_W−1(Ω),

where we have used Proposition 3.8. Since Ω⁺is a bounded Lipschitz domain, there exists a continuous linear operator E from W^k(Ω⁺) into W^k(Cⁿ), for anyk ≥0, such that for anyg∈W^k(Ω⁺),

Eg|_Ω+=g.

First extend f⁺ from Ω⁺ to Ef⁺ componentwise on D such that the following estimate holds,

kEf⁺k²_W1/2(D)≤Ckf⁺k²_W1/2(Ω⁺)

(such an extension exists using [21, Theorem 1.4.3.1]). In fact, one can chooseEf⁺ so that

kEf⁺k²_Wk(D)≤Ckf⁺k²_Wk(Ω⁺)

for allk. For our purposes, it suffices to know that V =

(−? ∂N ? ∂Ef⁺ on Ω,

0 onD\Ω,

defines a form satisfying ∂V = ∂Ef⁺ on Cⁿ and V is supported in Ω. Because the Cauchy-Riemann equations are not affected by forms involvingdz, the estimate in Proposition 3.8 is easily applied to (n, s)-forms. By applying the dual forms of these estimates, one obtains

kVk_W−1/2(Ω)≤εk∂Ef⁺k²_W−1/2(Ω)+C_εk∂Ef⁺k²_W−1(Ω)

≤εkf⁺k²_W1/2(Ω)+Cεkf⁺k².

Let ˜f⁺ = Ef⁺−V so that we have a ∂-closed form on all of Cⁿ that satisfies f˜⁺|_D\Ω=f⁺ and

kf˜⁺k²_W−1/2(Ω).εkfk²_L2(bΩ)+Cεkf⁺k².

Set u⁺ = ∂^∗N^Df˜⁺, where N^D denotes the ∂-Neumann operator for the ball D.

If we pick χ ∈C₀^∞(D) such that χ ≡1 on some neighborhood of Ω, we may use interior regularity to obtain

kχu⁺k²_W1/2(Ω).kf˜⁺k²_W−1/2(Ω). OnbΩ, one defines u=u⁻−u⁺. Thus∂bu=f and

kuk²_L2(bΩ) .kχu⁺k²_W1/2(Ω)+ku⁻k²_W1/2(Ω)

≤εkfk²_L2(bΩ)+Cεkf⁺k²+Cεkf⁻k².

Since kf⁺k_W1/2(Ω) and kf⁻k_W1/2(Ω) are both bounded by kfk_L2(bΩ) and k.k is compact with respect tok · k_W1/2(Ω) by the Rellich lemma, the result follows.

5. Proof of Theorems 1.2 and 1.3

The proof of the regularity in the Sobolev space W_r,s^k (Ω) of the Bergman pro- jectionB and the canonical solution operator∂^∗N for the∂-equation is the same as in Berndtsson-Charpentier [3].

Lemma 5.1. Let Ω⊂Cⁿ be a bounded Lipschitz q-pseudoconvex domain and let 1≤q≤n. Let δ(z) =−ρ(z), where ρis C²-defining function for Ω. Then, if we taking φ_β = −βlogδ, where β ∈(0,1) and u is any form which is orthogonal to L²_r,s−1(Ω, e^−φβ)∩ker∂,q+ 1≤s≤n−1, one obtains usuch that

Z

Ω

|u|²e^−φβdV ≤ Z

Ω

|∂u|²_i∂∂φ

βe^−φβdV. (5.1)

Proof. By using (1.1) and by taking φ=−klogδ, wherek is a positive constant, there existsα∈(0,1) such that (−δ^α) is strictly plurisubharmonic in Ω and

i∂φ∧∂φ <k α

i∂∂φ, on Ω.

Consequently, forσ≡1, one obtains from (3.1), kuk²_φ≤ k∂ uk²_φ+k∂^∗_φuk²_φ,

for any u ∈ C_r,s^∞(Ω) ∩dom∂^∗_φ. Thus, by the same argument of [11, Theorem 4.3.4], for q+ 1≤s ≤n−1, for every f ∈ L²_r,s(Ω, φ) with ∂f = 0, one can find u∈L²_r,s−1(Ω, φ) satisfies∂u=f and

Z

Ω

|u|²e^−φdV ≤c Z

Ω

|∂u|²e^−φdV. (5.2)

One can always select the solution u of (5.2) satisfying the additional property u∈L²_r,s−1(Ω, e^−φ)∩(ker∂)^⊥, i.e., satisfies

Z

Ω

e^{−φ t}u∧?υ= 0, (5.3)

for any∂-closed formυ∈L²_r,s−1(Ω, e^−φ). Hence, if we takingφ_β=−βlogδ, where β ∈ (0,1) andu is any form which is orthogonal to L²_r,s−1(Ω, e^−φβ)∩ker∂, one obtainsusuch that

Z

Ω

|u|²e^−φβdV ≤ Z

Ω

|∂u|²_i∂∂φ

βe^−φβdV.

Proposition 5.2. LetΩ⊂Cⁿ be a bounded Lipschitzq-pseudoconvex domain and let1≤q≤n. Letu=∂^∗_βN^βf be the solution to the equation∂u=f inL²_r,s(Ω, δ^β).

Then, by takingψk=−klogδ,k∈(0,1), forf ∈L²_r,s(Ω, δ^β−k),q+ 1≤s≤n−1, with∂f = 0, there exists a constantC1>0such that

Z

Ω

|u|²δ^β−kdV ≤C1

Z

Ω

|f|²_i∂∂(ψ

k+φβ)δ^β−kdV. (5.4) Proof. Sincef ∈L²_r,s(Ω, δ^β), thus by (5.2) there is a solution u∈L²_r,s−1(Ω, δ^β)∩ (ker∂)^⊥. Putg=ue^ψk=u δ^−k. Then

Z

Ω

|u|²δ^β−kdV = Z

Ω

|g|²δ^β+kdV. (5.5)

Thus, from (5.3), one obtains 0 =

Z

Ω

e^{−φβ t}u∧?υ= Z

Ω

e^{−(ψk+φβ)t}g∧?υ

= Z

Ω

δ^{β+k t}g∧?υ.

Thus, g is orthogonal to all ∂-closed forms of L²_r,s−1(Ω, δ^β+k), so by (5.1) one obtains

Z

Ω

|g|²δ^β+kdV ≤ Z

Ω

|∂g|²_i∂∂(ψ

k+φ_β)δ^β+kdV.

Thus, from (5.5), one obtains Z

Ω

|u|²δ^β−kdV ≤ Z

Ω

|∂g|²_i∂∂(ψ

k+φ_β)δ^β+kdV. (5.6)

Since, for any two real numbersaandb, and for everyε >0, one obtains 2|a| |b| ≤ε|a|²+1

ε|b|²,

and since∂g=δ^−k∂u+δ^−k∂ψ_k∧u. Thus, from (5.6), one obtains Z

Ω

|u|²δ^β−kdV ≤ Z

Ω

|∂u+∂ψk∧u|²_i∂∂(ψ

k+φ_β)δ^β−kdV

≤ Z

Ω

|∂u|²_i∂∂(ψ

k+φ_β)δ^β−kdV + Z

Ω

|∂ψk∧u|²_i∂∂(ψ

k+φ_β)δ^β−kdV + 2

Z

Ω

|∂u|_i∂∂(ψ

k+φ_β)|∂ψk∧u|_i∂∂(ψ

k+φ_β)δ^β−kdV

≤ 1 + 1

ε Z

Ω

|f|²_i∂∂(ψ

k+φβ)δ^β−kdV + (1 +ε)

Z

Ω

|∂ψk∧u|²_i∂∂(ψ

k+φβ)δ^β−kdV.

Since

i∂ψk∧∂ψk < t i ∂∂ψk

is valid for 0< t <1, the norm of the form∂ψk, measured in the metric with K¨ahler formi∂∂ψk is smaller thantat any point. Also, we can improve the estimate (5.1) by replacing|f|_i∂∂φ

βe^−φβby|f|_i∂∂(ψ

k+φ_β)e^−φβwithout having to change the weight function fromφβ toψk+φβ. Thus

|∂ψk∧u|²_i∂∂(ψ

k+φ_β)≤ |∂ψk|²_i∂∂(ψ

k+φ_β)|u|²≤ |∂ψk|²_i∂∂ψ

k|u|²≤t|u|². (5.7) By choosingεsmall such that (1 +ε)t <1, one obtains

Z

Ω

|u|²δ^β−kdV ≤C1

Z

Ω

|f|²_i∂∂(ψ

k+φ_β)δ^β−kdV

withC₁= (1 +¹_ε)/[1−(1 +ε)t].

Proposition 5.3. Let Ω ⊂ Cⁿ be a bounded Lipschitz q-pseudoconvex domain and let 1 ≤ q ≤ n. Then, for q+ 1 ≤ s ≤ n−1, the Bergman projection B^β mapsL²_r,s(Ω, δ^β−k)boundedly to itself, and the operator∂^∗_βN^β mapsL²_r,s(Ω, δ^β−k) boundedly to itself.

Proof. From the Kohn’s formula, one obtains

B^β=Id−∂^∗_βN_r,s+1^β ∂. (5.8)

Then, foru∈L²_r,s(Ω, δ^β−k) and forf ∈L²_r,s(Ω, δ^β−k)∩ker∂, one obtains hB^βu, fiβ,Ω=hu−∂^∗_βN^β∂u, fiβ,Ω

=hu, fiβ,Ω− h∂^∗_βN^β∂u, fiβ,Ω

=hδ^−ku, fiβ+k,Ω

=hδ^−ku, fiβ+k,Ω− h∂^∗_β+kN^β+k∂(δ^−ku), fiβ+k,Ω

=h(I−∂^∗_β+kN^β+k∂)(δ^−ku), fi_β+k,Ω

=hB^β+k(δ^−ku), fiβ+k,Ω

=hδ^kB^β+k(δ^−ku), fi_β,Ω.

Thus

B^β(δ^kB^β+k(δ^−ku)) =B^βu.

Using (5.8), one obtains

B^βu=B^β(δ^kB^β+k(δ^−ku))

= (I−∂^∗_βN^β∂)δ^kB^β+k(δ^−ku)

=δ^kB^β+k(δ^−ku)−∂^∗_βN^β(∂δ^k∧B^β+k(δ^−ku))

=δ^kB^β+k(δ^−ku)−k ∂^∗_βN^β∂δ

δ ∧δ^kB^β+k(δ^−ku) ,

(5.9)

because∂ B^β+k = 0.

For simplicity, write ξ = δ^kB^β+k(δ^−ku), for u ∈ L²_r,s(Ω, δ^β−k). Then, one obtains

Z

Ω

|ξ|²δ^β−kdV = Z

Ω

|δ^kB^β+k(δ^−ku)|²δ^β−kdV

= Z

Ω

|B^β+k(δ^−ku)|²δ^β+kdV

≤ Z

Ω

|δ^−ku|²δ^β+kdV

= Z

Ω

|u|²δ^β−kdV.

(5.10)

Thus, from (5.4), one obtains Z

Ω

∂^∗_βN^β(∂ψk∧ξ)

2δ^β−kdV ≤C1

Z

Ω

|∂ψk∧ξ|²_i∂∂(ψ

k+φ_β)δ^β−kdV. (5.11) From (5.7), one obtains

|∂ψk∧ξ|²_i∂∂(ψ

k+φ_β)≤ |∂ψk∧ξ|²_i∂∂ψ

k≤t|ξ|². (5.12) Substituting (5.10) and (5.12) into (5.11), one obtains

Z

Ω

|∂^∗_βN^β(∂ψ_k∧ξ)|²δ^β−kdV ≤C₁t Z

Ω

|u|²δ^β−kdV. (5.13) Thus, by using (5.9), (5.10) and (5.13), one obtains

kB^βuk²_β−k,Ω≤C2kuk²_β−k,Ω. (5.14) Thus, the Bergman projection B^β maps L²_r,s(Ω, δ^β−k) boundedly to itself. Since B^βu = (I−∂^∗_βN^β∂)uand ∂^∗_βN^βu =N^β∂^∗_βu, then ∂^∗_βN^βu = ∂^∗_βN^βB^βu and we already know thatB^β is bounded onL²_r,s(Ω, δ^β−k) we may as well assume from the start that∂f = 0. Then, by using (5.4) and (5.14), one obtains

k∂^∗_βN^βuk²_β−k,Ω=k∂^∗_βN^βB^βuk²_β−k,Ω≤C₁kB^βuk²_β−k,Ω≤C₁C₂kuk²_β−k,Ω. Thus, the operator∂^∗_βN^β mapsL²_r,s(Ω, δ^β−k) boundedly to itself.

Proposition 5.4. LetΩ⊂Cⁿ be a bounded Lipschitzq-pseudoconvex domain and let 1 ≤q≤n. Then, for k0 ∈ (0,1), the Bergman projection B and the operator

∂^∗N are exact regular inW_r,s^k (Ω) for0< k < k₀/2 and forq+ 1≤s≤n−1.