Estimates of fundamental solution for Kohn Laplacian in Besov and Triebel-Lizorkin spaces

ABSTRACT

We introduce Besov space ${\dot{B}}_p^{\alpha ,q}\left(\mathrm{\partial }{\mathrm{\Omega }}_k\right)$ and Triebel-Lizorkin space ${\dot{F}}_p^{\alpha ,q}\left(\mathrm{\partial }{\mathrm{\Omega }}_k\right)$ on a family of model domains $\mathrm{\partial }{\mathrm{\Omega }}_k=\left\{\right(\mathbf{z},z_{n+1})=(z_1,z_2,\dots ,z_{n+1}):\text{Im}\left(z_{n+1}\right)=\varphi \left(|\mathbf{z}{|}^2\right)\}$ with $\varphi \left(x\right)=x^k$ in ${\mathbf{C}}^{n+1}$ which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on $\mathrm{\partial }{\mathrm{\Omega }}_k$ in these spaces.

KEYWORDS:

• Fundamental solution
• Kohn Laplacian
• Besov space
• Triebel-Lizorkin space

MSC:

• 42B20
• 42B35
• 32A50

Disclosure statement

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

Additional information

Funding

The second author is partially supported by a McDevitt Endowment Fund at Georgetown University and the fourth author is partially supported by a NSF of China (Nos. 12071473 & 11671397).