$L^p$-estimates for elliptic pseudodifferential operators

Question

Assume we have an pseudodifferential operator

$P:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S}(\mathbb{R}^n), Pf(x) = (2\pi)^{-n/2}\int\mathrm{d}\xi\; p(x,\xi)\,\hat{f}(\xi)e^{i\xi x}$

acting on Schwarz-functions. Assume it is of degree $m$, that is we have $|\partial_\xi^\alpha \partial_x^\beta p(x,\xi)| \leq C_{\alpha,\beta} \, (1 + |\xi|)^{m - |\alpha|}$ for some real $m$ and multiindices $\alpha,\beta$. Now, assume that P is elliptic, that is $c(1+|\xi|)^m \leq |p(x,\xi)|$ for some constant.

Now it is a standard result, that one has the following elliptic $L^2$-estimate:

$\lVert f \rVert_{W^{s,2}} \le C (\lVert Pf\rVert_{W^{s-m,2}} + \lVert f\rVert_{W^{s-m,2}})$

My question: Is there a $L^p$-generalization of this estimate? If yes, where can I find a reference to it?

Answer 1

Yes you do have a generalization of your elliptic inequality to the $L^p$ case for $p\in (1,+\infty)$. In fact the operators with symbols in the class $S^0_{1,0}$ (as in your question) are bounded on $L^p$ ($p$ in the same range as above). You can find a proof of this in the Astérisque book by R. Coifman & Y. Meyer "Au delà des opérateurs pseudo-différentiels".