Meda Inequality for Rearrangements of the Convolution on the Heisenberg Group and Some Applications
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Research Article Meda Inequality for Rearrangements of the Convolution on the Heisenberg Group and Some Applications 2 ¨ V. S. Guliyev,1 A. Serbetci,2 E. Guner, and S. Balcı3 1
Department of Mathematical Analysis, Institute of Mathematics and Mechanics, AZ1145 Baku, Azerbaijan 2 Department of Mathematics, Ankara University, 06100 Ankara, Turkey 3 Department of Mathematics, Istanbul Aydin University, 34295 Istanbul, Turkey Correspondence should be addressed to A. Serbetci, [email protected] Received 13 May 2008; Revised 6 January 2009; Accepted 24 February 2009 Recommended by Yeol Je Cho The Meda inequality for rearrangements of the convolution operator on the Heisenberg group Hn is proved. By using the Meda inequality, an O’Neil-type inequality for the convolution is obtained. As applications of these results, some sufficient and necessary conditions for the boundedness of the fractional maximal operator MΩ,α and fractional integral operator IΩ,α with rough kernels in the spaces Lp Hn are found. Finally, we give some comments on the extension of our results to the case of homogeneous groups. Copyright q 2009 V. S. Guliyev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Let Hn Cn × R be the 2n 1-dimensional Heisenberg group. We define the Lebesgue space on Hn by Lp Hn
⎧ ⎨
f : f p ≡
⎩
1/p
Hn
fup du
⎫ ⎬ 0
1 r Q−α
|Ωv|f v−1 u dv,
1.3
Be,r
and the fractional integral with rough kernel by IΩ,α fu
Ωv −1 f v u dv. Q−α Hn |v|
1.4
It is clear that, when Ω ≡ 1, MΩ,α and IΩ,α are the usual fractional maximal operator Mα and the Riesz potential Iα see, e.g., 1–5 , respectively. In this paper, we obtain the Meda inequality for rearrangements of the convolution defined on the Heisenberg group Hn . By using this inequality we get an O’Neil-type inequality for the convolution on Hn . As applications of these inequalities, we find the necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and fractional integral operator with rough kernels from the spaces Lp Hn to Lq Hn , 1 < p < q < ∞, and from the spaces L1 Hn to the weak spaces WLq Hn , 1 < q < ∞. We also show that the conditions on the parameters ensuring the boundedness cannot be weakened for the fractional maximal operator and fractional integral operator with rough kernels. Finally, we extend our results to the case of homogeneous groups.
2. Preliminaries Let Hn be the 2n 1-dimensional Heisenberg group. That is, Hn Cn × R, with multiplication z, t · w, s z w, t s 2Imz · w,
2.1
where z · w nj1 zj wj . The inverse element of u z, t is u−1 −z, −t, and we write the identity of Hn as e 0, 0. The Heisenberg group is a connected, simply connected nilpotent Lie group. We define one-parameter dilatio
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