Fatou and littlewood theorems for poisson integrals with respect to non-integrable kernels

Abstract

Sjögren and others studied the boundary behavior of fractional Poisson integrals with respect to the fractional power of the Poisson kernel. We extend the fractional power of the Poisson kernel to a non-integrable kernel and investigate the boundary behavior of associated Poisson integrals. The existence of certain tangential limit (Fatou type theorem) as well as its sharpness (Littlewood type theorem) are given. The admissible tangency varies according to the integrability of the boundary function. Our Littlewood type theorem is new even for the fractional power of the Poisson kernel.

Keywords:

• Poisson integral
• Fatou theorem
• Littlewood theorem
• Tangential limit
• Approach region
• Non-integrable kernel
• AMS Subject Classifications: 31B25
• 31A20

Acknowledgement

The author would like to thank Peter Sjögren for useful comments. This work was supported in part Grant-in-Aid for (B) (No. 15340046) and Exploratory Research (No. 13874023) Japan Society for the Promotion of Science.

Notes

E-mail: [email protected]

Dedicated to the memory of Professor Matts Essén.

Dedicated to the memory of Professor Matts Essén.

Additional information

Notes on contributors

Hiroaki Aikawa †

E-mail: [email protected] Dedicated to the memory of Professor Matts Essén.