An inequality of Gårding on a manifold with boundary (Q1277406)

Let \(\gamma\geq 1\) be a real parameter and \(\Sigma^\mu\) be the class of symbols \(a\) with large parameter such that \[ | \partial_x^\alpha\partial_\xi^\beta a(x,\xi,\gamma)| \leq C_{\alpha,\beta}(\gamma^2+ | \xi| ^2)^{(\mu-| \beta|)/2} \] for all multi-indices \(\alpha\) and \(\beta\in{\mathbb{N}}^n\) with constants \(C_{\alpha,\beta}\) independent of \((x,\xi,\gamma)\in{\mathbb{R}}^n\times{\mathbb{R}}^n\times[1,\infty[\). Similarly, let \(\| u\| _\mu^2=\int(\gamma^2+| \xi| ^2)^\mu| \hat u(\xi)| ^2 d\xi\) and the \(L^2({\mathbb{R}}^n)\) scalar product \((u,v)=\int u(x)\overline{v(x)} dx\). Finally, let the operator \(a^w\) result from the Weyl quantification \[ a^wu(x)=(2\pi)^{-n}\int e^{i(x-y,\xi)}a\bigl((x+y)/2,\xi,\gamma\bigr)u(y) dy d\xi. \] The authors establish the following theorem: Let \(a\in\Sigma^1\) be a nonnegative symbol on\break \({\mathbb{R}}^n\times{\mathbb{R}}^n\times[1,\infty[\). Then there exists a seminorm \(C\) of \(a\) such that \(\bigl((x_n a)^wu,u\bigr)+C\| u\| _0^2\geq 0\), for all \(u\in C_0^\infty({\mathbb{R}}^n)\) with support contained in \(F=\{x\in{\mathbb{R}}^n;0\leq x_n\leq 4\}\) for all \(\gamma\geq 1\). The use of a second microlocalization is the main novelty of this paper. This theorem is used to prove the inequality Re\((a(x,D)u,u)\geq-C\| u\| _m^2\) for functions \(u\in S({\mathbb{R}}^n)\) supported in the half-space \(\{x_n\geq 0\}\) under the weak nonnegativeness condition Re \(a=b+x_n c\), where \(b\) and \(c\) are (everywhere) nonnegative symbols of odd integer order \(2m+1\). This result is used to improve uniqueness results in the linear Cauchy problem by means of Carleman's inequalities.