Compact Laplace-type bundle operators: regularity and shifted parametrices

Choose finitely many relatively compact bundle charts and subordinate smooth functions ${\rho }_i$ with sum one and supports strictly inside the charts. Each coefficient transformation on a compact overlap is multiplication by a smooth matrix followed by a smooth coordinate diffeomorphism with bounded derivatives and inverse derivatives at every fixed order. For a smooth coefficient function, the repeated chain rule expresses each derivative of the transformed function of order at most $k$ as a finite sum of bounded coefficients times derivatives of the original function of order at most $k$; change of variables bounds the integrals because the Jacobian and its inverse are bounded there. Approximate compactly supported Sobolev coefficients by [F1] smooth functions and pass these bounds and the weak derivative identities to the limit. Thus the same bounds hold on Sobolev classes. Multiplication by transition matrices is bounded by [F1]. Insert $u={\sum }_i{\rho }_{i}u$ to compare any two finite-chart norms; the finite sum yields bounds in both directions. The metric and density are bounded above and below on these compact supports, giving $H^0=L^2$ and equivalent norms.

Completeness is checked on the finitely many coefficient functions and derivatives: a Cauchy sequence has $L^2$ limits for every derivative of order at most $k$; passing the test-function identities to the limit identifies these as derivatives of the zeroth-order limit. The transition identities pass in $L^2$ by the bounds of step 1.1, so these local limits are one global section. This proves completeness for the sum-of-integral-derivatives inner product. To obtain smooth density, use cutoffs equal to one near each $\mathrm{supp}{\rho }_i$, mollify the zero-extended coefficient functions, multiply again by these cutoffs, extend the resulting sections by zero and sum; [F1] makes the sum converge in $H^k$. For a bounded $H^{k+1}$ sequence, all derivatives of the finitely many localized coefficients of order at most $k$ are bounded in $H^1$ on a fixed Euclidean ball. Apply [F3] successively to these finitely many families and retain a common subsequence converging in $L^2$ for every derivative. It is Cauchy in $H^k$, hence convergent. Apply the embedding of [F3] on the same balls and step 1.1 to get the $C^r$ bound when $k>r+n\mathrm{/}2$.

In a precompact chart each component of $Pu=f$ can be written $-{\mathrm{\partial }}_i(a^{ij}{\mathrm{\partial }}_j{u}_{\alpha })=f_{\alpha }-{\tilde{B}}_{\alpha \beta }^j{\mathrm{\partial }}_j{u}_{\beta }-C_{\alpha \beta }{u}_{\beta }\mathrm{.}$ The adjusted matrices $\tilde{B}$ include the derivatives of $a$. If $u\in H^1$, the right side is locally $L^2$ and bounded by $C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})$ on an outer patch. The weak scalar equation follows by testing the original distributional equation against a compactly supported test in the single component and integrating its principal part once. [F2] therefore gives all components in $H^2$ on an inner patch, with the same bound. Cover $M$ by finitely many inner patches, each with a larger precompact outer patch, to get $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})\mathrm{.}$ The lower matrices have been moved to the source; no coupled principal system or Dirichlet theorem is imported.

After step 2.2, interpolation [F2] on the finitely many zero-extended localized coefficients gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le \epsilon \mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}+C_{\epsilon }\mathrm{\parallel }u{\mathrm{\parallel }}_2$; cutoff derivatives add only lower-order terms, absorbed by choosing their interpolation tolerances small. Choose $C\epsilon <1\mathrm{/}2$ to obtain the stated $k=0$ estimate. For induction, suppose $f\in H^k$ and $u\in H^{k+1}$, $k\ge 1$. On nested patches differentiate the scalar principal equation $k$ times. For $\mathrm{\mid }\alpha \mathrm{\mid }=k$, $w={\mathrm{\partial }}^{\alpha }{u}_{\beta }\in H^1$ satisfies the same scalar second-order equation with source ${\mathrm{\partial }}^{\alpha }{f}_{\beta }$ plus terms involving at most $k+1$ derivatives of $u$: in the commutator with the principal term at least one derivative hits $a$, and the lower terms have order at most one. All these terms are $L^2$, with bound $C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+1}})$. Apply [F2] to $w$ on successively smaller patches and sum to get $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}}\le C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+1}})$. Repeated use of the interpolation inequality of [F2] gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+1}}\le \epsilon \mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}}+C_{k,\epsilon }\mathrm{\parallel }u{\mathrm{\parallel }}_2$; absorb again. Starting with step 2.2 and using $f\in H^k\subset H^{k-1}$ proves the required membership before this absorption. The embedding of step 2.1 then proves smoothness for smooth $f$.

Let $a(u,v)$ be the form obtained from $(Pu,v{)}_{L^2}$ by integrating the second-order principal term once, initially for smooth sections. It is bounded on $H^1\times H^1$: its integrands have at most one derivative in either factor and bounded coefficients. Formal symmetry and smooth density give $a(u,v)=\overline{a(v,u)}$. To prove a global lower bound, choose real smooth cutoffs ${\chi }_i$ supported in the charts with ${\sum }_i{\chi }_i^2=1$ (normalize a finite family of bumps by the positive smooth square root of their sum of squares). For smooth $u$, the principal integral of $a({\chi }_{i}u,{\chi }_{i}u)$ is bounded below by $c_i\mathrm{\parallel }\mathrm{\partial }({\chi }_{i}u){\mathrm{\parallel }}_2^2$: positive definiteness of the scalar coefficient and the positive bundle metric give this bound uniformly on its support. Every remaining term is bounded by $C\mathrm{\parallel }{\chi }_{i}u{\mathrm{\parallel }}_2\mathrm{\parallel }{\chi }_{i}u{\mathrm{\parallel }}_{H^1}+C\mathrm{\parallel }{\chi }_{i}u{\mathrm{\parallel }}_2^2$ and Young's inequality absorbs half the leading gradient. In the sum over $i$, $a(u,u)-{\sum }_{i}a({\chi }_{i}u,{\chi }_{i}u)$ contains only terms with a derivative on a cutoff; second derivatives on $u$ cancel by ${\sum }_i{\chi }_i^2=1$, and these remaining terms are bounded by $C\mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}+C\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. Absorb once more. Norm equivalence yields constants $c>0,C$ with $a(u,u)\ge c\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2-C\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\mathrm{.}$ Density extends this real inequality and the bounded form to all $H^1$.

Fix $\lambda >C+1$. Then $a_{\lambda }=a+\lambda (\cdot ,\cdot {)}_2$ is coercive on $H^1$, so [F4] gives for every $f\in L^2$ a unique $u=Bf\in H^1$ with $a_{\lambda }(u,v)=(f,v{)}_2$ for every $v\in H^1$, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le C^{\mathrm{\prime }}\mathrm{\parallel }f{\mathrm{\parallel }}_2$. Testing smoothly gives $Pu=f-\lambda u$ distributionally; step 3.1 gives $u\in H^2$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}\le C^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\parallel }f{\mathrm{\parallel }}_2$. Conversely any $u\in H^2$ has $Pu\in L^2$ and integration by parts, passed by smooth approximation, gives the same weak identity with $f=(P+\lambda )u$. Uniqueness shows both inverse identities and $\mathrm{ran}B=H^2$. Symmetry gives $(Bf,g{)}_2=a_{\lambda }(Bf,Bg)=(f,Bg{)}_2$, so $B$ is bounded selfadjoint. Its range contains all smooth sections and is therefore dense; $Bf=0$ implies $(f,v{)}_2=0$ for every smooth $v$, hence $f=0$. Compactness follows from its bounded $H^2$ range and step 2.1.

Selfadjointness of the unbounded domain is proved directly. Put $A=P+\lambda$ with domain $H^2=\mathrm{ran}B$. It is densely defined and symmetric. If $y$ belongs to $\mathrm{dom}{A}^{\ast }$ with $A^{\ast }y=z$, then $(Au,y{)}_2=(u,z{)}_2$ for every $u\in H^2$. Set $u=Bf$ for arbitrary $f\in L^2$ and use symmetry of $B$ to obtain $(f,y{)}_2=(Bf,z{)}_2=(f,Bz{)}_2$; thus $y=Bz\in H^2$ and $Ay=z$. Conversely symmetry puts every $H^2$ element in $\mathrm{dom}{A}^{\ast }$. Hence $A=A^{\ast }$, and subtracting the bounded real scalar $\lambda$ gives $P=P^{\ast }$ on exactly $H^2$. Higher regularity step 3.1, with $Pu=f-\lambda u$, inductively yields $B:H^k\to H^{k+2}$ bounded (use $\mathrm{\parallel }Bf{\mathrm{\parallel }}_2\le C\mathrm{\parallel }f{\mathrm{\parallel }}_2$ and the already obtained lower-order bounds), and gives smooth preservation.

The inverse identities in step 4.1 give $PB=I-\lambda B$ on $L^2$ and $BP=I-\lambda B$ on $H^2$. On $L^2$ the error is compact by step 4.1; on $H^2$ the restriction $B:H^2\to H^4$ is bounded by step 5.1 and $H^4\hookrightarrow H^2$ compact by two uses of step 2.1. Thus both are parametrices modulo compact operators on their respective spaces. All four conclusions, including the explicitly proved selfadjoint domain, follow.