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Boundedness of homogeneous fractional integrals on Lp for N/α ≤ P ≤ ∞

Published online by Cambridge University Press:  22 January 2016

Yong Ding  and
Shanzhen Lu
Yong Ding
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing, 100875, P.R.of China, dingy@bnu.edu.cn
Shanzhen Lu
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing, 100875, P.R.of China. lusz@bnu.edu.cn
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Abstract

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In this paper we study the map properties of the homogeneous fractional integral operator TΩ, α on Lp(ℝn) for n/α ≤ p ≤ ∞.

We prove that if Ω satisfies some smoothness conditions on Sn−1 then TΩ, α is bounded from Ln/α(ℝn) to BMO(ℝn), and from Lp(ℝn) (n/α < p ≤ ∞) to a class of the Campanato spaces l, λ (ℝn), respectively. As the corollary of the results above, we show that when Ω satisfies some smoothness conditions on Sn−1 the homogeneous fractional integral operator TΩ, α is also bounded from Hp(ℝn) (n/(n + α) ≤ p ≤ 1) to Lq(ℝn) for 1/q = 1/p-α/n. The results are the extensions of Stein-Weiss (for p = 1) and Taibleson-Weiss’s (for n/(n + α) ≤ p < 1) results on the boundedness of the Riesz potential operator Iα on the Hardy spaces Hp(ℝn).

Keywords

42B2047G10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 167 , 2002 , pp. 17 - 33
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

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