CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 265–285, August 2021 DOI: 10.4067/S0719-06462021000200265 On Rellich’s Lemma, the Poincaré inequality, and Friedrichs extension of an operator on complex spaces Chia-chi Tung 1 Pier Domenico Lamberti 2 1 Department of Mathematics and Statistics, Minnesota State University, Mankato Mankato, MN 56001, USA (emeritus). imggtn14@outlook.com 2 Dipartimento di Tecnica e Gestione dei Sistemi Industriali (DTG), University of Padova, Stradella S. Nicola 3-36100 Vicenza, Italy. lamberti@math.unipd.it ABSTRACT This paper is mainly concerned with: (i) a generalization of the Rellich’s Lemma to a Riemann subdomain of a complex space, (ii) the Poincaré inequality, and (iii) Friedrichs ex- tension of a Schrödinger type operator. Applications to the eigenfunction expansion problem associated to the modified Laplacian are also given. RESUMEN Este art́ıculo se enfoca principalmente en: (i) una general- ización del Lema de Rellich a un subdominio de Riemann de un espacio complejo, (ii) la desigualdad de Poincaré, y (iii) la extensión de Friedrichs de un operador de tipo Schrödinger. Se entregan también aplicaciones al problema de expansión de funciones propias asociado al Laplaciano modificado. Keywords and Phrases: Weighted Sobolev-Schrödinger product, Friedrichs extension, resolvent mapping. 2020 AMS Mathematics Subject Classification: 46E35, 26D10, 35J10, 32C30, 35J25. Accepted: 01 June, 2021 Received: 16 January, 2021 c©2021 C. Tung et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200265 https://orcid.org/0000-0003-0939-0714 https://orcid.org/0000-0003-2502-5661 266 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) 1 Introduction A milestone of pure and applied analysis since the last century is a selection principle of F. Rellich ([15], [4, p. 414]): Given a family of C 1-functions f in a bounded domain Ω ⊂ Rn with smooth boundary such that both the functions f and their first partial derivatives are uniformly bounded in the L2(Ω)-norm, then {f} contains a Cauchy subsequence with respect to the L2(Ω)-norm. One consequence of this far reaching result is the Rellich’s Principle: The Laplacian with zero boundary condition on a bounded domain Ω ⊂ RN has a compact resolvent. Thus the eigenfunctions of the equation −∆ψ = µψ in Ω, ψ|∂Ω = 0, form a complete orthogonal basis of L2(Ω). In view of some recurring interest concerning the Rellich embedding property on non-flat domains ([14], [13, 3.9.3], [11]), it seems of value to consider the question as to on what general domain the Rellich embedding and its consequences will remain valid. A main purpose of this note is to propose a setting of Riemann subdomains (in a complex space) on which an affirmative answer could be sought. In view of the fact that on a singular space analytic tools are not yet sufficiently developed 1, the basic notions of the Sobolev spaces on a Riemann subdomain (see [20]) need be properly formulated (to be recalled below). Assume throughout this paper that (D,p) is a (relatively compact) Riemann subdomain in a complex space Y of dimension m, meaning that D is a relatively compact open subset of Y and (as a subspace of Y ) admits a holomorphic map p : D → Cm with discrete fibers. Note that the pair (D,p) is a Riemann domain in the sense of [3, p. 19] and [8, p. 135]. If in addition p is a local homeomorphism, then (D,p) is said to be unramified. Every complex space of pure dimension m admits locally an open neighborhood (of each given point) and a finite, open holomorphic map onto a domain in Cm ([9, pp. 107-108]), hence is locally realizable as a Riemann subdomain. Let h denote a (2m + 1)-tuple (h0,h1, · · · ,h2m) of locally integrable functions hj on D. Set V := −h0, and denote by h the (2m + 1)-tuple obtained by replacing the initial entry by 0 (thus h = (0,h1, · · · ,h2m)). Let µ be a non-negative constant. A (2m + 1)-tuple h (as above) is called an allowable weight for (D,µ) if each component hj, j = 1, · · · ,m, is positive a. e. on D, and either (µ,V ) = (0,0) or c 0 := µ − ess sup D V > 0. (For convenience) call a 2m-tuple h′ = (h1, · · · ,h2m) with positive a. e., locally integrable components an allowable weight on D, and set h′ −→ := (0,h′). (Unless explicitly specified) in the following let h denote an allowable weight for (D,µ). Let D∗ be the largest open subset of D on which the map p = (p1, · · · ,pm) is locally biholomorphic, and set x̃j := Re(pj), ỹj := Im(pj), and 1 See Ruppenthal [17, p. 7]) “· · · whereas geometric and algebraic methods · · · are very well developed on singular spaces, most analytic tools are still missing”. CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 267 ∂j := ∂ ∂x̃k , j = 2k − 1, ∂j := ∂ ∂ỹk , j = 2k, 1 ≤ k ≤ m, on D∗. The space C 1(D) is equipped with the (weighted) Sobolev-Schrödinger product 〈w, v〉 µ,h := ((µ − V )w,v̄) D + [w,v] D,(h1,··· ,h2m) ∀w, v ∈ C 1(D), (1.1) where [ , ] D,h′ denotes the (weighted) Dirichlet product [w,v] D,(h1,··· ,h2m) := m∑ k=1 ∫ D∗ ( h2k−1 ∂w ∂x̃k ∂v̄ ∂x̃k + h2k ∂w ∂ỹk ∂v̄ ∂ỹk ) dυ̃. (1.2) Here dυ̃ denotes the pull-back of the Euclidean volume form on Cm under p. Clearly for all g ∈ C 1(D) one has ‖g‖2 µ,h ≥ ∫ D (µ − V )|g|2 dυ̃ ≥ c 0 ‖g‖2 L2(D) . The sesqilinear form “〈 , 〉 µ,h ” being elliptic, defines a scalar product on C 1(D). With respect to the induced norm ‖ ‖ µ,h the completion of C 1(D) gives rise to the Sobolev space H1µ,h(D), with induced scalar product (denoted by the same) and induced norm ‖ ‖ µ,h . Similarly the completion of the space C ∞,c(D) of test functions defines the Sobolev space H1µ,h,c(D) (see [20] and also [19]). Observe also that the Dirichlet product [w, v] D,(h1,··· ,h2m) = 〈w, v〉 1,(0,h1,··· ,h2m) − (w, v̄) D where w, v ∈ C 1,c(D) (given explicitly by the equation (1.2)) extends to a scalar product on H1 0,(0,h1,··· ,h2m) (D). The norm of f ∈ H11,(0,h1,··· ,h2m)(D) is given by ‖f‖ 1,(0,h1,··· ,h2m) = (∫ D |f|2 dυ̃ + [f,f] D,(h1,··· ,h2m) )1 2 . As an application of the inner products (1.1)-(1.2), an explicit representation of the Friedrichs ex- tension of the weighted Schrödinger operator 2 on a Riemann subdomain, allowing possibly singular points, is derived (Theorem 3.1). It is well-known that the employment of the Poincaré inequality plays a central role in the study of compact Sobolev embeddings. The connection between the (classical) Poincaré inequality and the Rellich embedding theorem (for Euclidean domains) was clarified by Galaz-Fontes [7]. 3 Given an allowable 2m-weight h′ on a non-flat subdomain (D,p), questions remain open, however, as to 2 This operator is motivated by the classical Schrödinger operator (appearing in the Schrödinger equation); the latter has been considered as “one of the most interesting objects in mathematical physics · · · ” (see [12, p. 3]). 3 The Poincaré inequality is sometimes referred to as the Poincaré-Friedrichs inequality. An underlying incentive for this paper is provided by a motivational remark in [7] (where by the “Friedrichs inequality” is meant the “Poincaré inequality”), “an explicit connection between the “Friedrichs inequality” and the Rellichs theorem has not been reported” (at least for a Riemann subdomain). 268 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) whether (either of) the following properties holds: (1) (D,p) has the Poincaré property relative to h′, in the sense that there exists a constant C D,h′ such that the following Poincaré inequality (with variable coefficients) holds: ‖f‖ L2(D) ≤ C D,h′ [f,f] 1 2 D,h′ ∀f ∈ H11,(0,h′),c(D); (1.3) (2) (D,p) is a Rellich domain with respect to h′, namely the Rellich property, “the embedding H1 1,h′ −→ ,c (D) →֒ L2(D) is compact”. The Rellich embedding property will be proved in §4 (independently of the Poincaré inequality for constant allowable weight) only for the case where (D,p) satisfies the following conditions: (i) (D,p) is of finite volume, namely, ∫ D∗ dυ̃ < ∞, and (ii) (D,p) is of Sobolev type, that is, there exists a constant α > 2 such that, for all f ∈ H1 1,(0,1,··· ,1),c (D), ‖f‖Lα(D) ≤ Const. ‖f‖1,(0,1,··· ,1). (1.4) Every Euclidean domain in Cm is of Sobolev type ([6, Theorem 6, p. 270]). Other examples are provided by finitely quasiregular Riemann subdomains (see §4). Theorem 1.1. (Rellich embedding theorem): If (D,p) is of finite volume and of Sobolev type, then (D,p) is a Rellich domain with respect to any constant allowable 2m-weight on D. Theorem 1.2. If (D,p) is an unramified Riemann subdomain in a complex space and if p defines an étale covering 4 , then (D,p) has the Poincaré property relative to any allowable 2m-weight h′ on D with h D := min {ess inf D (hj) | 1 ≤ j ≤ 2m} > 0. The Rellich embedding property can be extended to sections of a vector bundle over a differentiable manifold (see [10, p. 88 and p. 93], [22, p. 111]). It would be interesting to characterize those Riemann subdomains of an m-dimensional complex space which can be realized as a Rellich domain with respect to some allowable 2m-weight. A form of the Poincaré inequality (with respect to a continuous allowable 2m-weight) holds as a consequence of the Rellich embedding property (see Proposition 4.3). For a Riemann subdomain D with the Poincaré property relative to an allowable 2m-weight h′, the defining norms of the Sobolev spaces H1 1,h′ −→ ,c (D) and H1 0,h′ −→ ,c (D) 5 are equivalent. On such a domain the inhomogeneous Dirichlet problem (for the Poisson equation) admits a weak solution (Corollary 4.5). As a further application, consider the following (inhomogeneous) Dirichlet problem (to be referred to as a Poisson problem): − 2m∑ j=1 ∂j(hj∂jψ) + αψ = g a.e. in D, ψ|dD = 0, (1.5) 4 Defined in §4. 5 This equivalence is sometimes called the Friederichs’s Lemma. The traditional notation for the Sobolev space H1 1,(0,1,··· ,1),c (D) is W 1,2 0 (D). CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 269 where g ∈ L2(D) 6 . A (weak) solution on D of this problem is taken in the following sense: ψ is an element in H1 1,h′ −→ ,c (D) satisfying the equation − 2m∑ j=1 ∂j(hj∂jψ) + αψ = g weakly in D, namely, ψ satisfies the functional equation [ψ,v] D,h′ + α(ψ, v̄) D = (g, v̄) D , ∀v ∈ C ∞,c(D). (1.6) To solve this problem one must also determine the eigenvalues λ := −α. To this end, an alternative but more expedient formulation of the above equation is available (see Lemma 6.2). The latter requires the use of an operator, to be called a resolvent map, which can be introduced as follows: for each allowable 2m-weight h′ on D (as above with respect to h′), there is a linear mapping R D,h′ : L2(D) → L2(D) with image in H1 1,h′ −→ ,c (D) defined by the equation [R D,h′ f,v] D,h′ = (f, v̄) D (1.7) for all (f,v) ∈ L2(D)×H1 1,h′ −→ ,c (D). On a Rellich domain this map (in the special case of R D,(1,··· ,1) ) is in fact a compact mapping: Theorem 1.3. For any relatively compact Rellich domain D in Y, the resolvent map R D,{1,··· ,1} : L2(D) → L2(D) defined by [R D,{1,··· ,1} g,v] D,{1,··· ,1} = (g, v̄) D , ∀(g,v) ∈ L2(D) × H11,(0,1,··· ,1),c(D), is a compact, self-adjoint mapping. If (D,p) is a Rellich domain of type h′, then the behavior of the solutions of the boundary- eigenvalue problem − 2m∑ j=1 ∂j(hj∂jψ) + αψ = 0 a.e. in D, ψ|dD = 0, (1.8) can be determined, and similarly for the case of the Poisson problem (1.5). For completeness this spectral analysis is carried out in §6. Given mathematical physicists’ interest in complexified space-time models 7 , it is hoped that the results of this paper may be of use in some applications. 6 Here (and in the following) dD denotes the (maximal) boundary manifold of Dreg in the manifold Yreg of simple points of Y, oriented towards the exterior of Dreg ([18, p. 218]). 7 See, e. g., Hansen, R. O. and Newman, E. T. A complex Minkowski approach to twistors, GRG Vol.6, No.4 (1975), 361-385. 270 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) 2 Preliminaries In what follows every complex space is assumed to be reduced and has a countable basis of topology. For the definition and basic properties of differential forms on a complex space, see [18, §4.1]. In particular, the exterior differentiation d, the operators ∂, ∂̄ and dc := (1/4πi)(∂ − ∂̄) are well- defined ([18, Chap. 4]). For an open subset G of Y, denote by C νk (G) the set of C-valued k-forms of class C ν on G, C νk,c(G) the subspace of compactly supported k-forms (dropping the degree if k = 0), with ν = β to mean locally bounded, ν = m measurable, and ν = λ locally Lipschitzian ([18, §4]). Similarly for C νk (G). A measurable function 8 g on Y is said to be locally integrable (g ∈ L1loc(Y )) provided the form g χ is locally integrable on Yreg for every 2m-form χ ∈ C 0 2m(Y ). Similarly define L1(Y ) and L2loc(Y ) (for the latter, the above gχ is replaced by |g| 2χ). Denote by ‖z‖ the Euclidean norm of z = (z1, · · · ,zm) ∈ C m, where zj = xj +iyj. Let the space Cm be oriented so that the form υm := (ddc‖z‖2)m is positive. Let B(r) denote the r-ball in Cm centered at the origin and B[1] = {z ∈ Cm | ‖z‖ ≤ 1}. Let p : Y → Cm be a holomorphic map. If S ⊆ Y, let S′ := p(S); and in particular write a′ = p(a). Set p[a] := p − a′, ∀a ∈ Y. Clearly the form υp := dd c‖p[a]‖2 = ( i 2π ) ∂∂̄ ‖p[a]‖2 (2.1) is non-negative and independent of a. Denote by dυ the Euclidean volume element of Cm and define dυ̃ := p∗(dυ) on Y. If (D,p) is a Riemann subdomain, then dυ̃ is a semivolume form on D and dυ̃ = cmπ m m! υmp , where cm := (−1) m(m−1) 2 . If f, ψ ∈ L1loc(D), set (f,ψ) D := ∫ D f ψ dυ̃, provided the integral exists. Each element f ∈ L1loc(D) gives rise naturally to a top-dimensional current, T = [f] : χ 7→ ∫ f ∧ χ, with induced functional (cf. [21]) defined by 〈[f], φ〉 := (−1) m(m−1) 2 πm m! ∫ fφυmp , ∀φ ∈ C ∞,c(D). 3 Friedrichs extension of a Schrödinger type operator Given an allowable (2m + 1)-weight h for (D,µ) and g ∈ L2(D), the antilinear functional v 7→ (g, v̄) D is well-defined and continuous on H1µ,h,c(D). By the Riesz representation theorem, there is a unique element w ∈ H1µ,h,c(D) satisfying the equation 〈w, v〉 µ,h = (g, v̄) D , ∀v ∈ H1µ,h,c(D), (3.1) 8A function g on Y is said to be measurable on Y is so is the restriction g|Yreg. CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 271 with ‖w‖ µ,h ≤ const. ‖g‖L2(D). The assignment g 7→ Gµ,hg := w defines a linear, continuous mapping G µ,h of L2(D) onto its image R µ,h (D) := G µ,h (L2(D)). Clearly one has 〈G µ,h g,v〉 µ,h = (g, v̄) D , ∀v ∈ H1µ,h,c(D). (3.2) Thus G µ,h may be regarded as a (weak) Green’s map for the weighted Sobolev-Schrödinger oper- ator. Observe that 〈G µ,h v, v〉 µ,h = ‖v‖2L2(D), ∀v ∈ H 1 µ,h,c(D), (3.3) and 〈G µ,h ψ,v〉 µ,h = 〈G µ,h v, ψ〉 µ,h = 〈ψ,G µ,h v〉 µ,h ∀(ψ,v) ∈ H1µ,h,c(D) × H 1 µ,h,c(D). By the equation (3.3) the Green’s map G µ,h : L2(D) → H1µ,h,c(D) is injective, and the inverse map F µ,h := G−1 µ,h : R µ,h (D) → L2(D) is well-defined. Also, it is positive definite since given g ∈ L2(D), the equation (3.2) implies that (F µ,h (G µ,h g), G µ,h g) D = (g, G µ,h g) D = 〈G µ,h g,G µ,h g〉 µ,h = ‖G µ,h g‖2 µ,h ≥ const.‖G µ,h g‖2L2(D). For an allowable (2m + 1)-weight h for (D,µ) of class C 1 (namely, so is each component hj of h on D), define the Schrödinger operator S µ,h acting on C 1(D) (in the classical sense) by S µ,h ψ := (µ − V )ψ − 2m∑ j=1 ∂ j (hj ∂jψ) on D ∗. (3.4) Assume (in the following theorem) that h is an allowable (2m + 1)-weight for (D,µ) of class C 1. Define D µ,h := {ψ ∈ C 2,c(D)| S µ,h ψ ∈ L2(D)}, and H(D,S µ,h ) := H1µ,h,c(D) ∩ {w ∈ L2(D)| S µ,h [w] ∈ L2(D)} 9. Theorem 3.1. (Friedrichs extension of the operator S µ,h ) Let D be a Riemann subdomain in Y. Then the weighted Sobolev-Schrödinger operator S µ,h : D µ,h → L2(D) admits a positive, self- adjoint extension F µ,h : H(D,S µ,h ) → L2(D) with the property that for each w ∈ H(D,S µ,h ), (F µ,h w, v̄) D = 〈w,v〉 µ,h , ∀v ∈ H1µ,h,c(D), (3.5) and (F µ,h w, v̄) D = (w, S µ,h v̄) D , ∀v ∈ C ∞,c(D). (3.6) Proof. It will be shown that the inverted mapping F µ,h := G−1 µ,h acts as an extension of the operator S µ,h to R µ,h (D) = G µ,h (L2(D)). It follows from the symmetry of G−1 µ,h : L2(D) → R µ,h (D) that F µ,h : R µ,h (D) → L2(D) is self-adjoint, and satisfies the equation (F µ,h w, v̄) D = 〈w,v〉 µ,h , ∀(w,v) ∈ R µ,h (D) × H1µ,h,c(D). (3.7) 9“S µ,h [w]”] denotes the (weak) action of S µ,h on the functional [w]. 272 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) If further v is an element of C 2,c(D), then 〈w,v〉 µ,h = lim j→∞ 〈wj,v〉µ,h = lim j→∞ (wj, Sµ,hv̄)D. This implies that (F µ,h w, v̄) D = (w, S µ,h v̄) D . (3.8) To see that the mapping F µ,h is indeed defined on H(D,S µ,h ) one needs to check that R µ,h (D) = H(D,S µ,h ). Let w ∈ R µ,h (D). Clearly then w ∈ H1µ,h,c(D) and w = Gµ,hg for some g ∈ L 2(D). Then by the equation (3.8) one has, for all v ∈ C ∞,c(D), (w, S µ,h v) D = (ψ, v) D with ψ := F µ,h w ∈ L2(D), and thus S µ,h [w] = ψ as desired. Hence w ∈ H(D,S µ,h ). Conversely if w ∈ H(D,S µ,h ), then w ∈ H1µ,h,c(D) and Sµ,h[w] ∈ L 2(D). Thus there exists an element g ∈ L2(D) such that ( S µ,h [w], v ) D = (g, v) D , ∀v ∈ C ∞,c(D). For each v ∈ H1µ,h,c(D) and a sequence {vj} in C ∞,c(D) tending to v, 〈w,v〉 µ,h = lim j→∞ 〈w,vj〉µ,h = lim j→∞ (w, S µ,h v̄j)D = lim j→∞ (g, v̄j)D = (g, v̄)D, hence w satisfies the equation (3.2), thereby proving that w ∈ R µ,h (D). Therefore the formula (3.5) follows from the equation (3.7). Consequently the mapping F µ,h gives the Friedrichs exten- sion of S µ,h : D µ,h → L2(D). Remark 3.2. Especially the preceding theorem ensures that the existence of the Friedrichs exten- sion of the Laplacian 10 : −△{p} : Dµ,(1,··· ,1) → L 2(D). This extension is given by the mapping F : H(D,−△{p}) → L 2(D), and admits the representation (Fw, v̄) D = [w, v] D,(1,··· ,1) , ∀(w,v) ∈ H(D,−△{p}) × H 1 1,(0,1,··· ,1),c (D). In particular, for all (w,v) ∈ H(D,−△{p}) × C ∞,c(D), (Fw, v̄) D = (w, −△{p}v̄)D. 4 The Rellich Embedding Theorem and Poincaré inequality Let p : Y → Y ′ be a continuous mapping (between topological spaces). An open subset W ⊆ Y ′ is called a base domain evenly covered by p, if p−1(W) is a disjoint union of open subsets Bl ⊆ Y 10 Here “△{p}” denotes the (local) pullback to D ∗ under p of the Laplace operator of the Euclidean metric on Cm. CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 273 each of which is homeomorphic to W (under p); as such Bl will also be referred to as a covering sheet of p lying evenly over W (relative to p). If p : D → Cm is a light mapping 11 , then there exists an open connected neighborhood Ua ⋐ D of each a ∈ D, called a pseudoball at a ([21, §2]) such that, when restricted to the regular part Ûa := Ua\p −1(∆′) (∆ being the branch locus), the map p a = p : Ûa → U ′\ ∆′ is an unramified finite holomorphic covering, where U′ := B[a′](r) is an open ball in C m with center a′ (see [1, §2]). The sheet number of p a is equal to νp(a), the multiplicity of p at (the center) a ([ibid., §2]). Thus at each point z′ ∈ U′\∆′ there is an open ball Wz ′ ⋐ U′\∆′ (of radius < 1) (here z′ will be denoted by z′k and W z′ by Wk for reasons to become clear later), which is a base domain evenly covered by p a with covering sheets Bℓ = Bz ′ l ⊆ D, l = 1, · · · , l(k), over W k, each being (necessarily) biholomorphic to Wk under p. Here the number l(k) is equal to νp(a) (for each z ′ k) and Bkl has compact closure in Ûa. Definition 4.1. An admissible covering of a compact subset K of D∗ is an open covering of K consisting of open subsets Bℓ ⊆ D∗ with ℓ varying in a finite range such that each Bℓ is equal to some Bkl (namely some B z′k l ) with k in a finite range and l ∈ Z[1, l(k)]; moreover, for each fixed k the Bkl (with varying l) lie evenly (relative to p) over a base domain W k := Wz ′ k contained in some open ball Ŵk := Ŵz ′ k ⊂ Cm of volume < 1. Note that every bounded domain D in Cm admits an admissible covering (via a dilatation of the identity map). Also, if Ua is a pseudoball contained in D, then every compact subset K of Ûa admits an admissible covering. The set D∗ being σ-compact, one can write D∗ as a union of an exhausting (increasing) sequence {Kj} of compact subsets. Choose a C ∞-partition of unity {βj,ℓ} 1≤ℓ≤Nj on Kj subordinate to an admissible covering {Bj,ℓ} for Kj. One has, for each f ∈ C ∞,c(D), setting f{j} ℓ := (βj,ℓ) 1 2 f (the ℓ-th partitioned function of f relative to {Bj,ℓ}), ∫ Kj |f|2 dυ̃ ≤ Nj∑ ℓ=1 ∫ D∗ |f{j} ℓ |2 dυ̃. (4.1) A Riemann subdomain (D,p) in Y is said to be finitely quasiregular if (i) each compact subset K of D∗ admits a (finite) admissible covering {Bkl } with (corresponding) base domains W k with C 1-boundary contained in some open ball of finite radius in Cm; and (ii) the family {Wk} has finite cardinality K which depends only on D. If further D is unramified, then (D,p) is said to be finitely regular. The above definition of “finitely quasiregular domain” is equivalent to the following: “(D,p) is finitely quasiregular if the regular part D∗ is a finite union of local covering sheets B{ℓ}, ℓ = 11 A holommorphic mapping f : Y → Y ′ (between complex spaces) is light on D ⊆ Y if for each a ∈ D, dima f −1(f(a)) = 0 . 274 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) 1, · · · ,N”. For, the above definition implies that, given any (increasing) sequence {Kj} of compact exhaustion of D∗, the index sequence {Nj} (with Nj arising from a corresponding Kj) can be replaced by a finite sequence {1, · · · ,N} with N = l(1) + · · ·+ l(K). Thus D∗ is contained in (at most) N local covering sheets B{ℓ}. The converse assertion is trivially true. Note that every bounded domain in Cm is finitely regular. Also, each relatively compact subset D of the projective space Pm(C) contained in the open set Q{s}, s ∈ Z[1,m], is finitely regular relative to the Riemann covering p = p[s]. For a further example, consider the complex space Y = {(z,w) ∈ C2 |w2 = z3}. Relative to the projection (z,w) 7→ z the open set D := {(z,w) ∈ Y |‖z‖ ≤ r0} is a finitely quasiregular Riemann subdomain in Y (with two covering sheets). While for a general domain the Rellich Embedding Theorem may not be valid, the requirement that the domain be of Sobolev type is indeed somewhat restrictive. A characterization of the latter remains an open question. The following lemma is of some use: Lemma 4.2. Every finitely quasiregular Riemann subdomain (D,p) in Y is of finite volume and of Sobolev type. Proof. The regular part D∗ of a finitely quasiregular Riemann subdomain (D,p) admits a finite covering by local covering sheets Bℓ with ℓ varying in a finite range, say {1, · · · ,N}. It is easy to show that D has finite volume with respect to p. One can write D∗ as a union of an exhausting (increasing) sequence {Kj} of compact subsets. Each Kj admits a finite admissible covering {Bj,ℓ} 1≤ℓ≤Nj . The Sobolev Embedding Inequality on a bounded Euclidean domain ([6, Theorem 6, p. 270] (applied to each Bj,ℓ) implies that, for all g ∈ C ∞,c(D) 12, (‖g‖Lα(D)) α ≤ lim j→∞ Nj∑ ℓ=1 ∫ Bj,ℓ |g|α dυ̃ = lim j→∞ Nj∑ ℓ=1 (‖g‖Lα(Bj,ℓ)) α ≤ (C Bm )α lim j→∞ Nj∑ ℓ=1 (‖g‖H1 1,(0,1,··· ,1) (Bj,ℓ)) α ≤ (C Bm )α lim j→∞ Nj∑ ℓ=1 (‖g‖H1 1,(0,1,··· ,1) (D)) α for some constant C Bm (independent of D). Since the local covering sheets Bj,ℓ can be selected from the finite set {Bℓ}ℓ=1,··· ,N , the above index Nj can be replaced by N . Thus (‖g‖Lα(D)) α ≤ Const.(‖g‖H1 1,(0,1,··· ,1) (D)) α. The space C ∞,c(D) being dense in H1 1,(0,1,··· ,1),c (D), each element f therein can be approximated in the L2-norm by a sequence {g n } ⊂ C ∞,c(D). Thus the Sobolev inequality (1.4) follows. 12 For clarity denote the Sobolev norm of g ∈ H1 1,(0,1,··· ,1) (Bℓ) by ‖g‖ H1 1,(0,1,··· ,1) .(Bℓ) and similarly for g ∈ H1 1,(0,1,··· ,1),c (D). CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 275 Proof of Theorem. 1.1. For later reference, assume (unless otherwise mentioned) that h denotes a general allowable weight as indicated in the beginning of §1. Let {f n } be a bounded sequence in H1µ,h,c(D). It contains a subsequence {fnk } which converges weakly to some element f ∈ H1µ,h,c(D). By considering {fnk − f} it may be assumed that {fnk } ⊂ {fn} is a subsequence converging weakly to 0. In the following for simplicity denote this subsequence by the same notation {f n }. Since C ∞,c(D) is dense in H1µ,h,c(D), each fn is the limit of a sequence {φ n l } ⊂ C ∞,c(D) with respect to the ‖ ‖µ,h-norm on H 1 µ,h,c(D). Thus, for each n ∈ N there exists an element φ{n} ∈ C ∞,c(D) such that ‖f n − φ{n}‖µ,h < 1 n . Then for any given ρ ∈ H1µ,h,c(D), (φ{n},ρ)µ,h = (φ {n} − f n ,ρ)µ,h + (fn,ρ)µ,h → 0, n → ∞, namely φ{n} tends to 0 weakly in H1µ,h,c(D). Since ‖f n − φ{n}‖ L2(D) ≤ Const.‖f n − φ{n}‖µ,h, to prove the Rellich Embedding Theorem it suffices to consider the case where {f n } lies in C ∞,c(D) and converges weakly to 0 in H1µ,h,c(D) (in which case {fn} is uniformly bounded in H 1 µ,h,c(D)). In the rest of the proof, assume that (µ,V ) = (1,0). The goal is to show (possibly by passing to a subsequence) that limn→∞ ‖fn‖ 2 L2(D) exists and equals zero. For this purpose it will be necessary to swap the order of the limits in the following relation lim n→∞ lim j→∞ ∫ Kj |f n |2 dυ̃ = lim j→∞ lim n→∞ ∫ Kj |f n |2 dυ̃ (4.2) (where {Kj} is an increasing exhausting sequence of compact subsets of D ∗). This can be justified by verifying two conditions: (a) the sequence of functions Φ j (n) := ∫ Kj |f n |2 dυ̃ converges to Φ(n) := lim j→∞ ∫ Kj |f n |2 dυ̃ (as j → ∞) uniformly in n, and (b) for each fixed j the limit “limn→∞ ∫ Kj |f n |2 dυ̃” exists (in fact, equals 0). More explicitly, the assertion (a) amounts to showing that, for each ǫ > 0, ∃Jǫ and Nǫ ∈ N such that ∫ (KJǫ ) c |f n |2 dυ̃ ≤ ǫ, ∀n ≥ Nǫ. (4.3) To prove this condition, set 1 q′ := 2 α and 1 p′ := 1 − 1 q′ . Then by Hölder’s inequality (with f = 1 and g = |f n |2) the left-hand side of the above inequality is dominated by ∫ (KJǫ ) c |1 · |f n |2|dυ̃ ≤ (∫ (KJǫ ) c 1p ′ dυ̃ ) 1 p′ (∫ (KJǫ ) c (|f n |2)q ′ dυ̃ ) 1 q′ . (4.4) 276 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) Since (|f n |2)q ′ = |f n |α, and (‖|f n |2‖)Lq′ ((KJǫ )c) = (∫ (KJǫ ) c |f n |α dυ̃ ) 2 α = (‖f n ‖Lα((KJǫ )c)) 2, the inequality (4.4) becomes ∫ (KJǫ ) c |f n |2 dυ̃ ≤ |vol((KJǫ) c)|1− 2 α ‖f n ‖2Lα((KJǫ )c) . Therefore (if α is chosen to satisfy the Sobolev Inequality (1.4)) the second factor on the right-hand side is uniformly bounded, hence one has ∫ (KJǫ ) c |f n |2 dυ̃ ≤ Const. |vol((KJǫ) c)|1− 2 α . Finally, since (D,p) has finite volume, by letting Jǫ → ∞ one has vol((KJǫ) c) → 0. Thus the condition for uniform convergence (4.3) follows. By choice of an open covering of Kj by local covering sheets B ℓ (with notations as in Definition 4.1) each of which is contained in a fixed open neighborhood Dj ⋐ D ∗ (the sheets Bℓ and Dj being dependent only on j), all the terms f{j} n,ℓ have support contained in Dj. Denoting by f̂ {j} n,ℓ the direct image of f{j} n,ℓ |Bℓ (under the biholomorphic map p : Bℓ → Bℓ = p(Bℓ), the resulting family {f̂{j} n,ℓ } is uniformly bounded in H1µ,h,c(B ℓ) with respect to n, ℓ. Inscribe Bℓ in a cube C (independent of ℓ, and it may be assumed that C ⊂ B(1) by rescaling and translation). Every element ρ̂ ∈ C ∞,c(Bℓ) can be extended to C trivially (by setting ρ̂ to be zero off Bℓ). The weak convergence of {f n } (to 0) in H1µ,h,c(B ℓ) implies that of f{j} n,ℓ , hence f̂{j} n,ℓ converges weakly to 0 in H11,h,c(B ℓ). The preceding discussion shows that the compactness of the embedding “H11,h,c(D) →֒ L 2(D)” is a consequence of the following “claim”: If {f̂{j} n,ℓ } is a sequence (indexed by n) in H1 1,(0,h′),c (Bℓ) which converges weakly to 0 in H11,(0,h′),c(B ℓ), the h′ being an allowable constant 2m-tuple, then lim n→∞ ∫ D∗ |f{j} n,ℓ |2 dυ̃ = 0. Upon taking h = (0,h = ) with h = = (1, · · · ,1), this is an immediate consequence of the Rellich Embedding Theorem for a bounded Euclidean domain (here the open ball Bℓ). This proves the assertion (b). By interchanging the “n-” and the “j-limit” and making use of the relations (4.2) and (4.1), one has lim n→∞ lim j→∞ ∫ Kj |fn| 2 dυ̃ = lim j→∞ lim n→∞ ∫ Kj |fn| 2 dυ̃ ≤ lim j→∞ lim n→∞ Nj∑ ℓ=1 ∫ D∗ |f{j} n,ℓ |2 dυ̃ = 0. CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 277 Consequently Theorem 1.1 is proved in this case. Let h′ = (c1, · · · ,c2m) be a 2m-tuple of positive constants. Define p̂ = p̂ {h′} = (p̂ 1 , · · · , p̂ m ) : D → Cm by setting p̂ k (z) := x̂ k + iŷ k = (c 2k−1 )−1/2x̃ k + i(c 2k )−1/2ỹ k , 1 ≤ k ≤ m, Then the volume element dυ̂(h′) := p̂∗(dυ) = c −1/2 1 · · ·c −1/2 2m dυ̃. It is easy to show that the norm of an element f of Ĥ1 1,{0,1,··· ,1},c (D), that is, the space H11,{0,1,··· ,1},c(D) defined w. r. t. (D,p̂{h′}) with h ′ = (1, · · · ,1), is equivalent to the norm of f in H1 1,{0,c1,··· ,c2m},c (D) w. r. t. (D,p). Hence the Sobolev space Ĥ1 1,{0,1,··· ,1},c (D) is the same as H11,{0,c1,··· ,c2m},c(D). Since the embedding “ Ĥ 1 1,{0,1,··· ,1},c(D) →֒ L 2(D)” is compact, the same is true for the embedding “ H1 1,{0,c1,··· ,c2m},c (D) →֒ L2(D)”. This completes the proof Theorem 1.1. A Riemann subdomain (D,p) in Y is said to define a (distinguished) étale covering 13 (of p(D∗)) if (i) each compact subset K of D∗ admits a (finite) admissible covering {Bkl } (with corre- sponding base domains Wk); and (ii) the family {Wk} is pair-wise disjoint. Note that every bounded domain D in Cm defines a distinguished étale covering (via a rescaled identity map). As another example, consider the m-dimensional projective space Pm(C) and let Q{s} := {a = [a0, · · · ,am]|as 6= 0}, where s ∈ Z[1,m]. Then Q {s} can be regarded as an open Riemann subdo- main in Pm(C) relative to the s-th canonical coordinate map p[s] : Q{s} → Cm given by p[s] : a 7→ ( a0 as , · · · , âs as , · · · , am as ) (here “̂” denotes omission). Clearly every relatively compact subset D ⋐ Q{s} defines a distin- guished étale covering via the Riemann covering p = p[s]. Proof of Theorem 1.2. Let h′ = (h1, · · · ,h2m) be any allowable weight on D. To prove the Poincaré inequality (1.3), it suffices to verify it for all elements g ∈ C ∞,c(D). For, given f ∈ H1 1,h′ −→ ,c (D), there exists a sequence {gn} in C ∞,c(D) converging to f in H1 1,h′ −→ ,c (D), that is, (‖f − g n ‖2 L2(D) + [f − g n ,f − g n ] D,h′ ) 1 2 → 0 as n → ∞. (4.5) If ‖gn‖ L2(D) ≤ C D,h′ [gn,gn] 1 2 D,h′ , ∀n ≥ 1, then the limit relation (4.5) implies that inequality (1.3) holds. Thus the above claim holds true. Assume that (D,p) defines an étale covering {Bkl } of p(D ∗). Let g ∈ C 1,c(D) and K := Spt(g). The regular part D∗ being σ-compact, the set K∩D∗ admits an exhausting sequence of increasing compact subsets {Kj} of K ∩ D ∗, each of which is contained in a finite union of covering sheets 13 See Barlet [2, p. 110] for a closely related notion of “revêtement analytique étale”. 278 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) Bℓ (namely some Bkl as above). Choose a C ∞-partition of unity {βj,ℓ} 1≤ℓ≤Nj on Kj subordinate to such a covering {Bℓ}ℓ=1,··· ,Nj (with monotonically increasing index N j). Assume (without loss of generality) that ∑ 1≤ℓ≤Nj β j,ℓ = 1 on Dj := ∪{B ℓ| 1 ≤ ℓ ≤ Nj}. Let ĝ{j,ℓ} : Bℓ := p(Bℓ) → C be the direct image of g{j,ℓ} := βj,ℓg (under the map p). Assume now that D is unramified. Then D = D∗, hence the {Kj} can be chosen to be a finite sequence with Kj = K for large enough j. Then one has ∫ D |g|2 dυ̃ = ∫ Dj |g|2 dυ̃ = ∫ Dj ∑ ℓ (βj,ℓ)2 |g|2 dυ̃ = ∫ Dj ∑ ℓ (βj,ℓ)2 |g|2 dυ̃ + 2 ∫ Dj   ∑ ℓ 6=ℓ′ βj,ℓβj,ℓ ′   |g|2 dυ̃ = ∑ ℓ ∫ Bℓ |ĝ{j,ℓ}|2 dυ where the added integral (in the second equality) vanishes since the base domains of the covering sheets Bℓ are pair-wise disjoint. Note that g = ∑ 1≤l≤Nj β j,{l}g on Dj, By resorting to the Poincaré inequality for the Euclidean unit ball, one has 14 , for a given g ∈ C ∞,c(D), h D (P B )2 ∑ ℓ ∫ Bℓ |ĝ{j,ℓ}|2 dυ ≤ h D ∑ ℓ ∫ Bℓ |∇ĝ{j,ℓ}|2 dυ ≤ ∫ Dj ∑ ℓ 2m∑ λ=1 hλ|∂λ(g {j,ℓ})|2 dυ̃ = ∫ Dj 2m∑ λ=1 hλ ∑ ℓ, ℓ′ ∂λ(g {j,ℓ})∂λ(ḡ {j,ℓ′})dυ̃ = ∫ Dj 2m∑ λ=1 hλ ( ∑ ℓ ∂λ(g {j,ℓ}) )( ∑ ℓ′ ∂λ(ḡ {j,ℓ′}) ) dυ̃ = ∫ Dj 2m∑ λ=1 hλ∂λ ( ∑ ℓ g{j,ℓ} ) ∂λ ( ∑ ℓ′ ḡ{j,ℓ ′} ) dυ̃ = [g,g] Dj ,(h1,··· ,h2m) = [g,g] D,(h1,··· ,h2m) , where h D = min {ess inf D (hj) | 1 ≤ j ≤ 2m} > 0, and PB denotes the Poincaré’s constant for the unit ball. Consequently the Poincaré inequality (1.3) follows. Proposition 4.3 (Generalized Poincaré-Wirtinger Inequality15). Assume that (D,p) is a Rie- mann subdomain such that, with respect to a continuous allowable weight h′ on D, the Rellich- Kondrachov embedding property holds: H1 1,h′ −→ (D) →֒ L2(D) is compact. Then there exists a con- stant C D such that ∥∥∥∥f − 1 vol(D) ∫ D f dυ̃ ∥∥∥∥ L2(D) ≤ C D [f,f] 1 2 D,h′ , ∀f ∈ H1 1,h′ −→ (D). (4.6) 14 On the right-side of the following, in the integral of the second inequality a sum of terms “ const ∫ Dj ∑ λ hλ ∑ ℓ 6=ℓ′ ∂λ(g {j,ℓ})∂λ(ḡ {j,ℓ′})dυ̃” may be added, since it vanishes owing to the fact that the base domains W k are pair-wise disjoint. 15 A version of this inequality is presented in Deny-Lions [5, (5.5), p.329]. CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 279 The proof (by way of reductio ad absurdum) is omitted. Definition 4.4. Let D ⊆ Y be a Riemann subdomain with dD 6= ∅, f ∈ Lip(∂D; C) 16 , and ϕ ∈ C 02m(D\W ), where W is a thin 17 analytic subset of D. A weak solution of the Dirichlet problem for the Poisson equation: ddcu ∧ υm−1p = ϕ in D\W , u|dD = f|dD (4.7) is an element u = w ∈ H1 1,(0,1,··· ,1) (D) such that w ≡ f mod (H1 1,(0,1,··· ,1),c (D)), and [w,v] D,(1,··· ,1) = (ϕ,v) D , ∀v ∈ H11,(0,1,··· ,1),c(D). Corollary 4.5. Let D ⊆ Y be a Riemann subdomain with dD 6= ∅. Assume that (D,p) has the Poincaré property relative to the unit 2m-weight (1, · · · ,1). Then for any f ∈ Lip(∂D; C) and ϕ ∈ C 02m(D\W ), W being thin analytic in D, the Dirichlet problem (4.7) admits a weak solution w ∈ H1 1,(0,1,··· ,1) (D). Proof. Consider the linear mapping T : H1 1,(0,1,··· ,1),c (D) → C defined by T(v) := (ϕ,v̄) D − [f̃,v] D,(1,··· ,1) , ∀v ∈ H11,(0,1,··· ,1),c(D). where f̃ is a Lipschitzian extension of f to a neighborhood of D. Then the Poincaré inequality (1.3) (with h′ = (1, · · · ,1)) implies that the Riesz representation theorem is applicable to the operator T. Therefore there exists an element w 0 ∈ H1 1,(0,1,··· ,1),c (D) such that T(v) = [w 0 ,v] D,(1,··· ,1) , ∀v ∈ H11,(0,1,··· ,1),c(D). Then w := w 0 + f̃ ∈ H1 1,(0,1,··· ,1) (D) is a weak solution to the Dirichlet problem (4.7). 5 The resolvent map for an inhomogeneous Dirichlet problem Let h′ be an allowable 2m-weight on D and H1 1,h′ −→ ,c (D) be equipped with the Dirichlet norm (defined by (1.2)). For fixed f ∈ L2(D), applying the Riesz’s representation theorem to the bounded anti-linear functional v 7→ (f, v̄) D on H1 1,h′ −→ ,c (D), yields an element w ∈ H1 1,h′ −→ ,c (D) such that (f, v̄) D = [w, v] D,h′ , ∀v ∈ H1 1,h′ −→ ,c (D). 16 f ∈ Lip(∂D; C) means that f is (locally) Lipschitzian in a neighborhood of ∂D. As such it admits a Lipschitzian extension f̃ to a neighborhood of D by invoking a partition of unity. 17 A subset T of an m-dimensional complex space Y is thin, if at each point a ∈ T there is an analytic subset A of dimension < m in an open neighborhood U ⊆ Y of a such that T ∩ U ⊆ A. 280 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) The association “f 7→ R D,h′ f = w” defines a continuous linear mapping R D,h′ : L2(D) → L2(D) with image in H1 1,h′ −→ ,c (D) satisfying the equation (1.7). Assume now that (D,p) has the Poincaré property relative to a bounded, C ∞ allowable 2m- weight h′ 18. Then the Poincaré inequality (1.3) implies that the Sobolev spaces H1 1,h′ −→ ,c (D) and H1 0,h′ −→ ,c (D) are defined by equivalent norms, hence can be naturally identified with each other. The mapping G0,h′ −→ : L2(D) → H1 1,h′ −→ ,c (D) (the latter being identified with H1 0,h′ −→ ,c (D)) is continuous, linear and satisfies, for each fixed ψ ∈ L2(D), the equation (ψ,v̄) D = 〈G0,h′ −→ ψ,v〉 0,(0,h′) = [G0,h′ −→ ψ, v] D,h′ , ∀v ∈ H1 1,h′ −→ ,c (D). Hence it follows that R D,h′ = G0,h′ −→ : L2(D) → H1 1,h′ −→ ,c (D). Also, since the mapping G0,h′ −→ is injective, so is the mapping R D,h′ . Thus R D,h′ can (justifiably) be called the resolvent map for the differential operator defined by the left-side of the equation (1.8). Proof of Theorem 1.3. It follows from the hermitian symmetry of the Dirichlet product that the mapping R D,h′ : L2(D) → L2(D) is self-adjoint. In the rest of this proof write “(µ,h)” for the pair (1,{0,1, · · · ,1}). The mapping R D,h′ is expressible as a composition of the (restricted) mapping R̃ D,h′ : L2(D) → H1µ,h,c(D) and the compact Rellich embedding i : H 1 µ,h,c(D) →֒ L 2(D). Consequently R D,h′ = i ◦ R̃ D,h′ is compact. Remark 5.1. The above (same) proof of the embedding Theorem 1.3 yields the following: Assume that the embedding H1µ,h,c(D) →֒ L 2(D) is compact for a (given) allowable weight h for (D,µ). Then there is defined a compact mapping Rµ D,h′ : L2(D) → L2(D) (in a way similar to that for R D,h′ = R1 D,h′ : L2(D) → L2(D)). 6 Solution of an Eigenvalue Problem Theorem 6.1. Assume that (D,p) is a Rellich subdomain with respect to h′ (the latter being a given allowable 2m-weight). Then: (a) there exists a non-decreasing, unbounded sequence {λ j } of positive real numbers such that the operator equation [u, v] D,h′ = λ(u, v̄)D ∀v ∈ C ∞,c(D), u|dD = 0, (6.1) admits a nontrivial solution u ∈ H1 1,h′ −→ ,c (D) precisely when λ is a member of the countable set {λ j }; 18 That is, each component hj of h ′ is bounded on D; similarly define “C ∞-allowable weight on D”. CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 281 (b) there exists an orthonormal basis {ψ k } of H1 1,h′ −→ ,c (D), consisting of eigenfunctions of R D,h′ , which is complete in both L2(D) and H1 1,h′ −→ ,c (D); furthermore, if h is of class C ∞ on D∗, then each ψ k ∈ H1 1,h′ −→ ,c (D) ∩ C ∞(D∗). Proof. By the Remark 5.1, the mapping R D,h′ : L2(D) → L2(D) is compact, and self-adjoint. Hence it has real eigenvalues {βj}j∈N of finite multiplicity (equal to the dimension of the eigenspace Ej := ker(RD,h′ − βjI)) may be arranged as a sequence |β1| ≥ |β2| ≥ · · · , with no point of accumulation except possibly the origin. The members of (distinct) Ej and Ej′ are mutually orthogonal. Since the mapping R D,h′ takes values in H1 1,h′ −→ ,c (D), for each j = 1, 2 · · · and φj ∈ Ej, the relation R D,h′ φj = βjφj holds and implies that φj belongs to H 1 1,h′ −→ ,c (D). Moreover, βj [φj, φk]D,h′ = [RD,h′ φj, φk]D,h′ = (φj, φ̄k)D, ∀j, k = 1, 2, · · · . This implies that each βj > 0. By orthonormalizing a basis of each Ej and taking their union, one obtains an orthonormal basis {ψ k } of H1 1,h′ −→ ,c (D) consisting of eigenfunctions of R D,h′ . One can arrange that each ψk has eigenvalue βk (by repeatedly listing the same βj as many times as its multiplicity, namely, dimEj). Thus ψk − 1 βk R D,h′ ψk = 0 ∀k = 1, 2 · · · , so that ψk is a solution to a Dirichlet Problem of the type (1.5) (with g = 0), which is equivalent to solving the functional equation (6.1) (with λk := 1 βk > 0 and λk ↑ ∞). To prove the completeness of the system {ψ k } in H1 1,h′ −→ ,c (D), recall the fact that ker(R D,h′ ) = {0}. The desired conclusion follows then from the completeness criterion of [16, p. 234]. By a standard regularity criterion, if h is of class C ∞ on D∗, then each eigenfunction ψ k belongs to C ∞(Q) for all (open) domains Q ⋐ D∗. Consequently ψ k belongs to H1 1,h′ −→ ,c (D) ∩ C ∞(D∗). Lemma 6.2. Given g ∈ L2(D), the Poisson problem (1.5) admits a solution ψ in H1 1,h′ −→ ,c (D) if and only if the following functional equation on C ∞,c(D), (I + α R D,h′ )ψ = w (6.2) with w := R D,h′ g ∈ H1 1,h′ −→ ,c (D), admits a solution ψ in H1 1,h′ −→ ,c (D). Proof. The above equation (1.6) can be expressed alternatively as a functional equation on C ∞,c(D) in the form [ψ,v] D,h′ + α [R D,h′ ψ,v] D,h′ = [w,v] D,h′ , where w := R D,h′ g ∈ H1 1,h′ −→ ,c (D). From this the equation (6.2) follows. 282 Chia-chi Tung & Pier Domenico Lamberti CUBO 23, 2 (2021) Theorem 6.3. Assume that (D,p) is a Rellich subdomain with respect to h′ = (1, · · · ,1). Let g ∈ L2(D). Consider the Poisson problem − △pψ + αψ = g a.e. in D, ψ|dD = 0. (6.3) (A) If α 6∈ {−λj}j=1,··· ,∞ (the λj being the eigenvalues of −△p), then there exists a unique weak solution ψ ∈ H1 1,(0,1,··· ,1),c (D) of the problem (6.3) with ‖ψ‖ 1,(0,1,··· ,1) ≤ Const.‖g‖L2(D). (B) If α = −λj (for some λj as above), then the problem (6.3) has a weak solution ψ ∈ H1 1,(0,1,··· ,1),c (D) if and only if (g, ψ̄) D = 0 for each ψ = ψkj , k = 1, · · · ,s, the latter being the associated eigenfunctions of the problem of Theorem 6.1: (I − λjRD,(1,··· ,1))(ψ) = 0. Each solution of the the inhomogeneous problem (6.3) is of the form ψ = ψ0 + s∑ k=1 ckψ k j , (6.4) where ψ0 is a fixed solution and the ck are suitable constants. Proof. In this proof let the allowable weight h be (0,(1, · · · ,1)). Since the Rellich embedding j : H1 1,(0,1,··· ,1),c (D) →֒ L2(D) is compact, so is the composition RD,(0,1,··· ,1) ◦j : H 1 1,(0,1,··· ,1),c (D) → H1 1,(0,1,··· ,1),c (D). It is known that for a constant α 6= −λj, the operator I + αRD,(0,1,··· ,1) ◦ j is invertible with a bounded inverse. Therefore the equation (6.2) has a unique solution ψ = (I + αR D,(0,1,··· ,1) )−1w1 with w1 := RD,(0,1,··· ,1)g ∈ H 1 1,h,c (D), and ‖ψ‖ 1,h ≤ ‖(I + αR D,(0,1,··· ,1) )−1‖ ‖w1‖1,h. Since ‖w1‖1,h ≤ Const.‖g‖L2(D), the assertion in (A) is proved. To prove the assertion in (B), observe that the closure of the range of an operator is the orthogonal complement of the null space of its adjoint. The equation ψ − λj RD,(0,1,··· ,1)ψ = v1, (6.5) where v 1 = R D,(0,1,··· ,1) g ∈ H1 1,h,c (D), has a solution if and only if v 1 ∈ R(I − λj RD,(0,1,··· ,1)), which is equivalent to v 1 ⊥ ker(I − λjR ∗ D,(0,1,··· ,1) ). The latter means that v is orthogonal (with respect to the inner product [ , ] D,(0,1,··· ,1) on H1 1,h,c (D)) to all the eigenfunction ψkj corresponding to the eigenvalue λj, namely, [v 1 ,ψkj ]D,(0,1,··· ,1) = (g, ψ̄ k j )D = 0. The expression (6.4) is a consequence of the equation (6.5). CUBO 23, 2 (2021) Poincaré inequality and Rellich’s Lemma 283 Acknowledgement The authors gratefully acknowledge the referee’s suggestions, which have led to improvements of this paper. Initial inquiries concerning this work were started during a sabbatical granted by Dean B. 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