6.1 Regularity of Operator
We will fix a manifold , which will be thought of as , and consider the manifold . We will write and . We let be a Hermitian vector bundle on .
Let be a first-order self-adjoint elliptic operator on , and consider the differential operator
This is the differential operator of interest on . While the definition resembles that of a heat operator, it may not be the best idea to think of it as a heat operator. We will later introduce a further time variable and consider , and confusion might arise.
The basic problem we want to solve is the equation
For now, we assume , and later extend to more general functions after we establish the right bounds.
As before, since is self-adjoint and elliptic, we can find an eigenbasis of . We write
The we see that for the equation to hold, we must have
We have previously decided that the boundary condition we want is for . There is another way we can think about this boundary condition. Observe that the solution to this differential equation is well defined up to adding . If , then there is at most one solution with reasonable growth as . However, if , then we can add any multiples of and have a sensible solution ( is somewhat of an edge case). Thus, fixing gives us a unique “sensible” solution.
For convenience, we write for the composition
where is restriction and the last map is the spectral projection onto the subspace spanned by the eigenvectors with non-negative eigenvalues. Then our boundary condition is . By an abuse of notation, we write for , and the boundary condition for will then by . We shall write for the space of smooth sections such that .
Imposing this boundary condition, we can write down explicit solutions for :
We check that for large , the function is either exponentially decreasing (if ), constant (if ) or identically zero (if ). The constant term wrecks our hope that this takes values in , but we can still hope it takes values in .
Note also that for , the value of depends on the future values of , which ties in with our previous point — at large , all future values of are zero, hence .
We let be the function that sends to above, which is clearly linear. The next proposition will, in particular, show that this map is well-defined, i.e. is actually smooth:
extends to a continuous map .
Recall that since is elliptic, we have
So we can equivalently define the norm by
Using that the eigenspace decomposition is orthogonal in and the defining equations
we get an inequality
To bound in terms of , we use the (rotated) Laplace transform
One checks easily that up to a constant which we omit, we have Parseval's identity:
We can explicitly compute
Using Parseval's identity, and the fact that
we obtain bounds
So if we ignore the term, would send into . The term is eventually constant, given by the integral of the compactly supported function . So maps into .
Using the equation
and calculating as above, we see that gets mapped into .
The main takeaway from the section is then the following theorem:
There is a linear operator
Moreover, extends to a continuous map for all integral . It is given by convolution with a kernel , where is now allowed to take negative values. The kernel is smooth away from .□