# ap.evaluation of pdes – Hipoellipticity or parabolic regularity for vector bundles Answer

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## ap.evaluation of pdes – Hipoellipticity or parabolic regularity for vector bundles

Let $$E to M$$ breathe a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not essentially compact). Let $$H : Gamma(E) to Gamma(E)$$ breathe a differential operator with flush coefficients such that its principal attribute is $$g$$, the Riemann tensor of $$M$$ (so $$H$$ is a generalized Laplacian). If $$p : (0, infty) instances M to M$$ is the unaffected projection, let $$F = p^* E$$ breathe the pull-back bundle. Let $$u$$ breathe a piece in $$F$$ with distributional values such that $$(partial_t – H) u = 0$$ within the distributional sense.

Are there ready-made instruments out there that will enable me to resolve that $$u$$ is flush on $$(0, infty) instances M$$?

The query reminds one of many ideas of hypoellipticity or parabolic regularity for capabilities. I might attempt to mimick these and bear analogous outcomes for sections, however for positive I’m not the primary one to necessity them, so possibly they’ve already been obtained and I simply do not know the place to search for them (a Google search did not ameliorate both). I too consider that this drawback is not only some brisk corollary of the same outcomes for capabilities.

(In my concrete drawback $$u$$ is a few $$L^2 _{loc}$$-integrable part. $$(0, infty)$$ might very properly breathe changed with $$mathbb R$$, of passage, however in my labor I necessity the warmth semigroup afterward.)

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