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Ellipticity and invertibility in the cone algebra on \(L_p\)-Sobolev spaces. (English) Zbl 0992.58010

Let \(B\) be an \((n+1)\)-dimensional manifold with conical singularities, \(\mathbb{B}\) the blow up of \(B\) obtained to replace a neighborhood of a conical singularity \(b\) by \([0,1)\times X_b\), where \(X_b\) is a smooth compact \(n\)-dimensional manifold, and the cross-section of the cone with the vertex \(b\) (thoughout the paper, \(B\) is assumed to have only one cone singularity). Fixing a density on \(\mathbb{B}\) and a boundary defining function \(t\), the natural \(L_p\)-space \(L_p(B)\) on \(B\) is defined to be the space of all measurable functions \(u\) on \(\mathbb{B}\) such that \(\int|u|^p t^n d\mu <\infty\).
In this paper, an ellipticity condition of a pseudodifferential operator \(A\) on \(\mathbb{B}\) is given in terms of its interior symbol and conormal symbol, and it is shown that the spectrum of \(A\) considered on \(L_p(B)\) (denoted by \(A_p)\) does not depend on \(p\) except for a discrete set without accumulation points. If \(A\) is an order zero operator, change of index of \(A_p\) according to \(p\) is computed to be \[ \text{ind} A_p- \text{ind} A_q =\sum_{(n+1)/q <{\mathfrak R}z<(n+1)/p} M\bigl( \sigma^0_M (A),z\bigr), \quad 1<p<q< \infty, \] (Theorem 3.20). Here \(M(h,z)\) is the multiplicity of \(h\) at \(z\) in the sense of I. C. Gokhberg and E. I. Sigal [Math. USSR, Sb. 13, 603-625 (1971); translation from Mat. Sb., Nov. Ser. 84(126), 607-629 (1971; Zbl 0254.47046)]. To show this, the authors use weighted Mellin \(L_p\)-Sobolev spaces \({\mathcal H}^{s,\gamma}_p (\mathbb{B})\), \(s,\gamma\in \mathbb{R}\), \(1<p <\infty\). For \(s\in \mathbb{N}\), this space is the set of \(u\in H^s_{p,\text{loc}} (\text{int } \mathbb{B})\) such that \[ t^{(n+1)/2- \gamma} (t\partial_t)^k \partial_x^\alpha u(t,x)\in L_p\left( {dt\over t} dx \right),\;\forall k+|\alpha|\leq s, \] and \(L_p(B)={\mathcal H}_p^{0,\gamma_p} (\mathbb{B}) \), \(\gamma_p=(n+1)/(1/2-1/p)\). Precise definitions of \({\mathcal H}_p^{s,\gamma} (\mathbb{B})\) are given in Sect. 2 as the completions of \(C^\infty_\alpha (\mathbb{B})\), the Fréchet space of those smooth functions on \(\text{int} \mathbb{B}\) whose divergence of all derivatives at \(b\) are suppresed by \(t^{(n+1)/2-\gamma} (1+ |\log t|^2)^{l/2}\), for all \(l\in\mathbb{N}\). The cone algebra \(C^\mu(\mathbb{B}; (\gamma,\gamma -\mu,\Theta))\) of order \(\mu\) is also defined in this Section as the space of continuous operators \(A:C^\infty_\gamma (\mathbb{B})\to C^\infty_{\gamma -\mu}(\mathbb{B})\) whose singularities at \(b\) are given by Mellin operators (Definition 2.8). After having defined the conormal symbol of a cone operator, the ellipticity condition of a cone operator is defined by the nonvanishing of the homogeneous principal symbol and invertibility of the conormal symbol (3.1).
The authors remark that this ellipticity condition seems to be stronger than Schulze’s condition [B. W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics; Math. Topics 4, Akademie Verlag, Berlin (1994; Zbl 0810.35175), Definition 1.2, p. 16], but they are the same (Proposition 3.16). By using properties of parameter dependent pseudodifferential operators explained in Section 1, it is shown if a cone operator is elliptic, then the kernel and index of its extension to the weighted Mellin \(L_p\)-Sobolev space are independent of \(s\) and \(p\) (Corollary 3.5).
It is also shown that a cone operator is invertible as a bounded operator \({\mathcal H}_p^{s,\gamma} (\mathbb{B})\to {\mathcal H}_p^{s-\mu, \gamma-\mu} (\mathbb{B})\) for some \(s\) and \(p\), then this is valid for all \(s\) and \(p\) (Corollary 3.15).
Since the Green operators are only defined for a fixed choice of weight data and the conormal symbol may have a pole, a priori \(A\) can not be considered on different \(L_p(B)\). But the authors construct a family of cone operators \(A_p:L_p(B)\to L_p(B)\) given by \[ A_p=\omega_0 \text{op}_M^{ \gamma_p-n/2} (h+h_0)\omega_1+(1-\omega_2) A_\psi(1- \omega_3). \] Here \(\omega_i\) are cutoff functions, \(h_0\) has no pole on the vertical line \(\Gamma_{(n+1)/p}\) and \(A_\psi\in L^0_{cl} (\text{int} \mathbb{B})\), where \(\Gamma_c= \{z\in\mathbb{C} \mid {\mathfrak R}z=c\}\). Then it is shown \(A_p\) and \(A_q\) coincide on \(C_0^\infty (\text{int} \mathbb{B})\) if and only if \(h_0\) has no singularity between the lines \(\Gamma_{(n+1)/p}\) and \(\Gamma_{(n+1)/q}\) (Lemma 3.18). By this Lemma, if \(A_p: L_p(B)\to L_p(B)\) is elliptic for some \(p\), then it is elliptic for all but finitely many \(1<p<\infty\) (Corollary 3.19). Theorem 3.20 follows from Corollaries 3.5 and 3.19. After giving an example of Theorem 3.20, the paper is concluded by showing that the set of all \(p\) such that \(A_p\) is invertible, is open (Theorem 3.22).

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
47L15 Operator algebras with symbol structure
47A53 (Semi-) Fredholm operators; index theories
35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
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