## Tokyo Journal of Mathematics

### On a Weak $L^1$ Property of Maximal Operators on Non-Compact Semisimple Lie Groups

#### Abstract

We shall give a simple proof of the weak type $L^1$ inequality for the $K$-bi-invariant Hardy-Littlewood maximal functions on non-compact real rank one semisimple Lie groups. For higher rank groups we do under an assumption which holds for the most parts. And on $SU(n,n+k)$ we introduce a maximal operator defined by the characteristic function supported on a cube, and show that the operator also satisfies the weak $L^1$ property.

#### Article information

Source
Tokyo J. Math., Volume 25, Number 1 (2002), 165-180.

Dates
First available in Project Euclid: 5 June 2009

https://projecteuclid.org/euclid.tjm/1244208943

Digital Object Identifier
doi:10.3836/tjm/1244208943

Mathematical Reviews number (MathSciNet)
MR1908220

Zentralblatt MATH identifier
1010.22014

#### Citation

KAWAZOE, Takeshi; LIU, Jianming. On a Weak $L^1$ Property of Maximal Operators on Non-Compact Semisimple Lie Groups. Tokyo J. Math. 25 (2002), no. 1, 165--180. doi:10.3836/tjm/1244208943. https://projecteuclid.org/euclid.tjm/1244208943

#### References

• J. L. Clerc and E. M. Stein, $L^p$-multipliers for non-compact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A., 71 (1974), 3911–3912.
• G. van Dijk and S. C. Hille, Canonical representations related to hyperbolic spaces, J. Funct. Anal., 147 (1997), 109–139.
• M. Flensted-Jensen and T. H. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat., 11 (1973), 245–262.
• G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes 28 (1982), Princeton University Press.
• R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, A Series of Mordern Surveys in Math 101 (1988), Springer.
• B. Hoogenboom, Spherical functions and invariant differential operators on complex Grassmann manifolds, Ark. Mat., 20 (1982), 69–85.
• T. Kawazoe, Atomic Hardy spaces on semisimple Lie groups, Japan. J. Math., 11 (1985), 293–343.
• T. Kawazoe, Hardy spaces and maximal operators on real rank one semisimple Lie groups I, Tohoku Math. J., 52 (2000), 1–18.
• J. Liu, Maximal functions associated with the Jacobi transform, Bull. London Math. Soc., 32 (2000), 1–7.
• J-O. Strömberg, Weak type $L^1$ estimates for maximal functions on non-compact symmetric spaces, Ann. of Math., 114 (1981), 115–126.