Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups

ABSTRACT

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the $(\rho ,\delta )$-Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case $\rho =\delta .$ Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated.

KEYWORDS:

• Sub-Laplacian
• compact Lie group
• Besov spaces
• Nikolskii's inequality

AMS SUBJECT CLASSIFICATIONS:

• Primary 22E30
• Secondary 58J40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A is defined by $f\mapsto \mathrm{Af}(x)={\int }_{{\mathbb{R}}^n}{e}^{i2\mathrm{\pi x}\cdot \xi }\sigma (x,\xi )\hat{f}(\xi )\mathrm{d}\xi ,$ $\hat{f}$ is the Fourier transform of f, and σ satisfies the $(\rho ,\delta )$-conditions $|{\mathrm{\partial }}_x^{\beta }{\mathrm{\partial }}_{\xi }^{\alpha }\sigma (x,\xi )|=O((1+|\xi |{)}^{-m-\rho |\alpha |+\delta |\beta |}).$

2 A is defined by $f\mapsto \mathrm{Af}(x)=\sum _{[\xi ]\in \hat{G}}{d}_{\xi }\mathrm{Tr}[\xi (x)\sigma (x,\xi )\hat{f}(\xi )],$ $\hat{f}$ is the Fourier transform of f, and σ satisfies the $(\rho ,\delta )$-conditions $\Vert {\mathrm{\partial }}_x^{\beta }{\mathrm{\Delta }}_{\xi }^{\alpha }\sigma (x,\xi ){\Vert }_{\mathrm{op}}=O(⟨\xi {⟩}^{-m-\rho |\alpha |+\delta |\beta |}).$

Additional information

Funding

The first author was supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations. The second author was supported in parts by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, EPSRC grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. This is a revised version of the submitted one to the arxiv as arXiv:1901.06825 on 21-January-2019