A Gårding Inequality on a Manifold with Boundary
Nicolás Lerner, Xavier Saint Raymond · Birkhäuser Boston eBooks · 2001
Let a ∈ C∞ (ℝ n × ℝ n ) and m ∈ ℝ. The function a is said to be a symbol of order m if one has estimates $$ \left| {\partial _x^\partial \partial _\xi ^\beta a\left( {x,\xi } \right)} \right| \leqslant C_{\alpha ,\beta } \left( {1 + \left| \xi \right|} \right)^{m - \left| \beta \right|} $$ uniformly on ℝ n × ℝ n for all multi-indices α, β ∈ ℤ + n . With such a symbol we associate the pseudo-differential operator (of order m) a(x, D): S(ℝ n ) → S(ℝ n ) defined by $$ a\left( {x,D} \right)u\left( x \right) = \left( {2\pi } \right)^{ - n} \int {e^{i\left\langle {x,\xi } \right\rangle } a\left( {x,\xi } \right)\hat u\left( \xi \right)d\xi } . $$.