• DOI:10.1007/S00208-015-1281-1
• Corpus ID: 62895015

Smoothing estimates for non-dispersive equations

@article{Ruzhansky2015SmoothingEF,
  title={Smoothing estimates for non-dispersive equations},
  author={Michael Ruzhansky and Mitsuru Sugimoto},
  journal={Mathematische Annalen},
  year={2015},
  volume={365},
  pages={241-269}
}

• Michael Ruzhansky, M. Sugimoto
• Published 3 August 2015
• Mathematics
• Mathematische Annalen

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 $$a(D_x)$$a(Dx) of order m satisfying the dispersiveness condition $$\nabla a(\xi )\ne 0$$∇a(ξ)≠0 for $$\xi \not =0$$ξ≠0, the global smoothing estimate $$\begin{aligned} {\left\| {{\left\langle {x}\right\rangle }^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} \varphi (x)}\right\| }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\| {\varphi }\right\| }_{L^2{\left( {{\mathbb R}^n_x}\right) }} \quad (s>1/2) \end{aligned}$$x-s|Dx|(m-1)/2eita(Dx)φ(x)L2Rt×Rxn≤CφL2Rxn(s>1/2)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 $$\begin{aligned} {\left\| {{\left\langle {x}\right\rangle }^{-s}|\nabla a(D_x)|^{1/2} e^{it a(D_x)}\varphi (x)}\right\| }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\| {\varphi }\right\| }_{L^2{\left( {{\mathbb R}^n_x}\right) }} \quad (s>1/2) \end{aligned}$$x-s|∇a(Dx)|1/2eita(Dx)φ(x)L2Rt×Rxn≤CφL2Rxn(s>1/2)which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $$a(D_x)$$a(Dx). We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators $$a(D_x)$$a(Dx), where $$\nabla a(\xi )$$∇a(ξ) may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.