COMMUNICATIONS ON doi:10.3934/cpaa.2020074 PURE AND APPLIED ANALYSIS Volume 19, Number 3, March 2020 pp. 1463{1483 FREDHOLM THEORY FOR AN ELLIPTIC DIFFERENTIAL OPERATOR DEFINED ON Rn AND ACTING ON GENERALIZED SOBOLEV SPACES Melvin Faierman School of Mathematics and Statistics, The University of New South Wales UNSW SYDNEY, Sydney, NSW 2052, Australia (Communicated by Yuri Latushkin) Abstract. We consider a spectral problem for an elliptic differential oper- n 0,χ n ator debined on R and acting on the generalized Sobolev space Wp (R ) n for 1 < p < 1. We note that similar problems, but with R replaced by n either a bounded region in R or by a closed manifold have been the subject of investigation by various authors. Then in this paper we establish, under the assumption of parameter-ellipticity, results pertaining to the existence and uniqueness of solutions of the spectral problem. Furthrermore, by utilizing the aforementioned results, we obain results pertaining to the spectral properties of the Banach space operator induced by the spectral problem. 1. Introduction. In his well known book [12], H¨ormanderintroduced in Chapter 2 a class of weight functions defined on Rn, which he denoted by K, and the Banach space Bp;k; k 2 K; 1 ≤ p ≤ 1, composed of tempered distributions u such that F u n n is a measurable function on R and kF u 2 Lp(R ), where F denotes the Fourier transformation on Rn. He then investigated various properties of this space as well as the regularity properties of solutions of partial differential operators acting on Bp;k. Since the space B2;k, called H¨ormanderspace when k is restricted to a certain subset of K, is of particular importance as it gives us a significant generalization of n classical Sobolev space based on L2(R ), the work of H¨ormanderhas in recent times stimulated significant investigations and to the establishment of important results pertaining to elliptic boundary for elliptic operators acting on the restriction of H¨ormanderspace to bounded regions in Rn and over closed manifolds, respectively (see for example [17, 18,5], as well as the book [19]). Shortly after the appearance of [12] there appeared the paper of Volevich and Paneyakh [24] presenting, by means of a certain subset K0 of weight functions in K, a generalization of classical Bessel-potential space for 1 < p < 1, which they µ denote by Hp ; µ 2 K0, and then decribe various properties of this space as well as µ n the spaces obtained by restricting the members of Hp to subsets of R . We have mentioned above that B2;k gives us a significant generalization of Sobolev n space based on L2(R ). Hence motivated by the works cited above, our aim in this paper is firstly to remove the restiction p = 2, and by fixing our attention upon 2000 Mathematics Subject Classification. Primary: 47A10, 47A53, 47B38, 47F05. Key words and phrases. Generalized Sobolev spaces, parameter-ellipticity, existence, unique- ness, spectral theory. 1463 1464 MELVIN FAIERMAN certain classes of weight functions in K, to introduce our generalization of classical n Sobolev space based on Lp(R ), 1 < p < 1, and then describe various properties of these spaces. Secondly, we are then going to appeal to the results of Cordes and his collaborators concerning the spectral theory for elliptic operators defind on Rn and acting on classical Sobolev spaces in order to establih various results concerning the spectral theory for a parameter-elliptic differential operator defined on Rn and acting on generalized Sobolev space (for the relevant literature concerning the works just cited we refer to [6,7,8, 13], and to the references listed therein). We might mention at this point that the works of Cordes and his collaborators just cited as well as an earlier work of the author [9] motivated by their work constitute a small part of the vast literature dealing with elliptic operators defined on Rn or on unbounded domains in Rn (we refer to the book [25] by Volpert for the relevant references). However, as far as we can ascertain, all of these works deal with operators acting on either Sobolev spaces, or on certain weighted Sobolev spaces, or on H¨olderspaces. Hence what is novel about this paper is that it is the first paper dealing with the spectral theory for elliptic operators defined on Rn and n acting on generalized Sobolev spaces based on Lp(R ), i.e., on spaces of H¨ormander type. Accordingly, in this paper we shall be concerned with the spectral problem n A(x; D)u(x) − λ u(x) = f(x) for x 2 R ; (1) P α n where n ≥ 2;A(x; D) = jαj ≤ 2m aα(x)D is a differential operator defined on R of order 2m and λ 2 L, where L is a closed sector in the complex plane with vertex at the origin. Our assumptions concerning the spectral problemn (1) will be made precise in Section3. In Section2 we make precise our definition of generalized Sobolev space over n χ n R . This is achieved by introducing the space Hp (R ) which is a generslization of n Lp(R ) and which is based upon a class of weight functions χ belonging to certain subsets of the H¨ormanderclass of weight functions K. We also present various χ n χ n properties of the space Hp (R ), and then extending the definition of Hp (R ), we introduce the generalized Sobolev spaces which will be considered in this paper, k,χ n 0,χ n χ n namely Hp (R ); k 2 N0, where Hp (R ) = Hp (R ) (see Definition 2.5 below). In Proposition5 we show the connection between the generalized Sobolev space k,χ n k n Wp (R ) and the calssical Sobolev space Wp (R ). In Section3 we make precise our assumptions concerning the spectral poroblem (1) as well as the definition of parameter-ellipticity. Then under the assumption of parameter-ellipticity we establish our main results pertaining to the existence and uniqueness of of solutions (1) (see Theorems 3.4 and 3.5 below). We might mention that these results are obtained by a consideration of a auxilliary spectral problem of the form (1), but with A(x; D) replaced by the differential operator A~(x; D), and with this auxilliary spectral problem posed in classical Sobolev spaces. Finally, in Section4 the results of Sections2 and3 are employed to derive our main results pertaining to the Fredholm properties of the Banach space operator χ χ 0,χ n Ap , where Ap denotes the operator acting on Wp (R ) which acts like A(x; D) 2m,χ n and has domain Wp (R ) (see Theorems 4.1 and 4.2 below). 2. Preliminaries. In this section we are going to introduce the generaliztion of classical Sobolev space considered in this paper and discuss some of its properties. FREDHOLM THEORY 1465 0 n Accordingly, we let x = (x1; : : : ; xn) = (x ; xn) denote a generic point in R α α1 αn and use the notion Dj = −i@=@ xj;D = (D1;:::;Dn);D = D1 ··· Dn and α α1 αn 0 n ξ = ξ1 ··· ξn for ξ = (ξ1; : : : ; ξn) = (ξ ; ξn) 2 R , where α = (α1; : : : αn) is a Pn 2 1=2 multi-index whose length j=1 αj is denoted by jαj. We also put h ξ i = (1+jξj ) and indicate differentiation with respect to another variable, say y 2 Rn, instead α α of x by replacing D and D by Dy and Dy , respectively. For 1 < p < 1; k 2 n k N0 = N [ f 0 g, and G an open set in R , we let Wp (G) denote the usual Sobolev space of order k related to Lp(G), and denote the norm in this space by k u kk;p;G = P R α p 1=p k ( jα|≤k G jD u(x)j dx) for u 2 Wp (G) unless otherwise stated, since at times k n in the sequel we shall also equip Wp (R ) with its equivalent Bessel-potential space norm (see [23, p.177]). Note that in the sequel it will always be supposed, unless otherwise stated, that 1 < p < 1, since at times we will also have to deal with k n the space W1 (R ). In addition we will use norms depending upon the parameter λ 2 C, namely for k 2 N0 with k ≤ 2m, we let k=2m jjjujjjk;p;G = k u kk;p;G + jλj k u k0;p;G k n for u 2 Wp (R ). We refer to [11] for details concerning parameter dependent norms. Furthermore, we let S(Rn) denote the Schwartz space of rapidely decreasing func- tions on Rn, let S0(Rn) denote its dual, and equip S0(Rn) with its weak* topology. Finally we let R± = f t 2 R t ? 0 g. ` n Next for ` 2 N0 we let C (R ) denote the vector space consisting of functins φ which, together with their partial derivatives of order up to `, are continuous on Rn (in the sequel we shall also introduce the space C1(Rn) which is defined analogously). In addition we let C`(Rn) denote the subspace of C`(Rn) consisting of all functions φ 2 C`(Rn) for which φ as well as its partial derivatives of order up to ` are bounded on Rn. Lastly for ` 2 N we let C`;0(Rn) denote the subspace of C`(Rn) consisting of functions φ for which Dαφ(x) ! 0 as jxj ! 1 for 1 ≤ jαj ≤ `, while for ` = 0 we let C0;0(Rn) denote the subspace of C0(Rn) consisting of functions n φ for which !φ(x) ! 0 as jxj ! 1, where !φ(x) = sup φ(x) − φ(y) for x; y 2 R , and where for each x, the supremum is taken over those y for which jx − yj ≤ 1 Note that for ` ≥ 1;C`;0(Rn) ⊂ C0;0(Rn).