# Smoothing estimates for non-dispersive equations

Mathematische Annalen, Jun 2016

This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors’ previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393–423, 2012), where dispersive equations were treated. For operators $$a(D_x)$$ of order m satisfying the dispersiveness condition $$\nabla a(\xi )\ne 0$$ for $$\xi \not =0$$, the global smoothing estimate \begin{aligned} {\left\| {{\left\langle {x}\right\rangle }^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} \varphi (x)}\right\| }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\| {\varphi }\right\| }_{L^2{\left( {{\mathbb R}^n_x}\right) }} \quad (s>1/2) \end{aligned} is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form \begin{aligned} {\left\| {{\left\langle {x}\right\rangle }^{-s}|\nabla a(D_x)|^{1/2} e^{it a(D_x)}\varphi (x)}\right\| }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\| {\varphi }\right\| }_{L^2{\left( {{\mathbb R}^n_x}\right) }} \quad (s>1/2) \end{aligned} which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $$a(D_x)$$. We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators $$a(D_x)$$, where $$\nabla a(\xi )$$ may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00208-015-1281-1.pdf

Michael Ruzhansky, Mitsuru Sugimoto. Smoothing estimates for non-dispersive equations, Mathematische Annalen, 2016, 241-269, DOI: 10.1007/s00208-015-1281-1