L1Estimates for Maximal Functions and Riesz Transform on Real Rank 1 Semisimple Lie Groups
✍ Scribed by Takeshi Kawazoe
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 481 KB
- Volume
- 157
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Let G be a real rank one semisimple Lie group and K a maximal compact subgroup of G. Radial maximal operators for suitable dilations, the heat and Poisson maximal operators, and the Riesz transform, which act on K-bi-invariant functions on G, satisfy the L p -norm inequalities for p>1 and a weak type L 1 estimate. In this paper, through the Fourier theories on R and G we shall duplicate the Hardy space H 1 (R) to a subspace H 1 s (G) (s 0) of L 1 (G) and show that these operators are bounded from H 1 s (G) to L 1 (G).
1998 Academic Press
where A is parametrized as [a x ; x # R] (for the definition of \ see (3) below). We let F 1 f (x)=e \x F f (x). Then the integral formula for the Iwasawa decomposition of G (cf. [6, p. 373]) yields that f # L 1 (G) if and only if
and thus, article no.