(?)-neumann Operator Regularity And Compactness

In several complex variables, a fundamental and general problem is to construct holomorphic functions, which have certain specified properties,on domains or manifolds. One way to construct such functions is by studying solutions to the Cauchy-Riemann equations (?)u=f (1) by patial differential equation methods. But in high dimension(more than one variable), Canchy-Riemann system is overdetermined. One method for dealing with this difficulty is by considering certain related, second order systems of partial differential equations. That is, the (?)-Neumann problem:The (?)-Neumann problem was formulated by D. C. Spencer in the fifties as a means to generalize the theory of harmonic integrals (i.e. Hodge theory) to non-compact complex manifolds. From the point of view of partial differential equations, the (?)-Neumann problem represents the prototype of a problem where the operator is elliptic. But the boundary conditions are not coercive ([43]), so that the classical elliptic theory does not apply. From the point of view of several complex variable, the importance of the problem is that its solution provides the solution of (?)-equation. In Chapter One, we introduce the (?)-Neumann problem, its boundary conditions, the existence of the (?)-Neumann problem in L² and the relationship between the solution of (2) and the solution of (?)-equation.In Chapter Two, the author mainly talks about the regularities of (?)-Neumann operator, there are two kinds of regularities of (?)-Neumann operator: (a). Global regularity: For any f∈C_(p,q)^∞((?)), there exists a smooth solution u∈C_(p,q-1)^∞(D) which satisfies (2). (b). Exact regularity: For any f∈W_(p,q)^s(D), u =(?)^*Nf, the canonical (Kohn) solution of (2) belongs to W_(p,q)^s(D)。It is obviously that exact regularity can imply global regularity by the Sobolev imbedding theory.Let D be a smooth bounded strictly pseudoconvex domain in Cⁿ. J. J. Kohn solved the (?)-Neumann problem on D ([39]), and showing the subelliptic estimate of the (?)-Neumann problem ([42]). Hormander ([36]) proved certain Carleman type estimates which in the case of bounded pseudoconvex domains imply the existence of N_q as a bounded self-adjoint operator on L². O. Abdelkader and S. Saber ([1, 2]) proved the elliptic estimate of the (?)-Neumann problem on the strongly pseudoconvex domains with the piecewise smooth boundary and the strongly pseudoconvex domains with the lipschitz boundary. H. P. Boas and E. J. Stranbe ([16]) proved the exact regularity of the (?)-Neumann problem for the smooth bounded pseudoconvex domains admitting the defining function that is plurisubharmonic on the boundary. Mei-Chi Shaw and J. Michel ([52]) get the exact regularity of the (?)-Neumann problem for the Lipschitz pseudoconvex domains with plurisubharmonic defining function.H. P. Boas and E. J. Straube ([14]) showed that for the complete Hartogs domain in C², the (?)-Neumann operator was exact regular. So-Chin Chen proved the exact regularity of the (?)-Neumann problem for transverse symmetric circular domain and the Reinhardt domain ([21, 22, 23]).To general smooth bounded (weakly) pseudoconvex domain, the global regularity does not always hold. Barrett ([4]) gave a counterexample: the worm domain. Kiselman ([38]) gave an example, a bounded pseudoconvex Hartogs domain but not smooth, on which the global regularity of the (?)-Neumann problem does not hold. Christ ([27]) proved that the prior estimate does not hold on the worm domain, or it was contrary to Barrett's result.A sufficient condition to the regularity of (?)-oNeumann operator is Condition T. And the vector method is to find a vector field satisfies Condition T.Condition T ([26]): (?)∈＞0, there exists a smooth real vector field T = T_∈, depending on∈, defined in some open neighborhood of D and tangent to the boundary with the following properties:(â…°). On the boundary, T can be expressed asfor some smooth function a_∈(z) with |a_∈(z)|≥δ> 0 for all (?)z∈bD, whereδis a positive constant independent of∈.(â…±).If S is any one of the vector fields L_n, (?)_n, L_jk, (?)_jk, 1≤j＜k≤n, thenfor some smooth function A_s(z) with sup |A_s(z)|＜∈. Hereand for 1≤j＜k≤n.The author considers the exact regularity of the (?)-Neumann problem for the domains which are invariant under certain S¹-group action. In the paper the author mainly considers the following four kinds S¹-action: andThe way is to find the vector fields which satisfy the Condition T respectively.To get the regularity of the (?)-Neumann, except the vector method introduced in chapter two, the compact estimate is another way since Kohn and Nirenberg ([43]) showed that the compactness of N_q implies the global regularity for the smooth bounded domains.As to compactness, we have the following lemma: Lernma 0.1.[35, 36] LetΩbe a bounded pseudoconvex domain, for 1≤q≤n, the following are equivalent:(i). (?)-Neumann operator N_q is compact from L_(0,q)²(Ω)to itself(ii). the embedding of the space D^0,1= {u∈A^0,1((?))|< u,(?)r > (z) = 0, z∈bD}, provided with the graph norm , into L_0,q²(Ω)is compact.(iii). For every∈> 0, there exists a constant C(∈) > 0, such that ||u||²â‰¤âˆˆQ(u,u)+ c(∈)||u||²-1where u∈Dom((?))n Dom((?)*), Q(u,v):= ((?)u,(?)u)+ ((?)^*u,(?)^*u),||·||_-1² is the Sobolev norm of order -1 on D.(iv). the canonical solution operators (?)*N_q : L_(0,q)²(Ω)→L₍(0,q-1)²(Ω) and (?)*N_q+1:L_0,q+1²(Ω)→L_0,q²(Ω) are compact.The statement in (iii) is called a compactness estimate. In (iv), we refer to (?)*N_q and (?)*N_q+1 as "canonical solution operators", although strictly speaking the solution operators are the restrictions to the kernel of (?). The equivalence of (ii) and (iii) is in [43], Lemma 1.1. That (i) is equivalent to (ii) and (iii) is an easy consequence of the general L²-theory and the fact that L_0,q²(Ω) embeds compactly into W_(0,q)^-1(Ω). And the equivalence of (i) and (iv) follows from the formula N_q = ((?)*N_q)*((?)*N_q) + ((?)*N_q+1)((?)*N_q+1)* ([34]).Catlin ([19, 20], also can see [17, 18, 35]) gave a sufficient condition of the compactness of the (?)-Neumann operator: Definition 0.1. For any M > 0, if there exists a functionφ=φM∈C²(D), such that(1). |φ|≤1 on D,(2). for any, p∈bD, and (?)∈Cⁿ, i(?)(?)φ(p)((?),(?))≥M||(?)||². then we say the boundary of D satisfies Property P.In chapter three, the author uses the "twisted estimate" of Ohsawa-Takegoshi ([54]) to prove the theorem of Catlin ([19]): Theorem 0.2. Let D be a bounded pseudoconvex domain in Cⁿ, 1≤q≤n. If the boundary of D satisfies the Property P, then the (?)-Neumann operator N_q is compact.