Multi-parameter fractional integration on Heisenberg group
Chuhan Sun, Zipeng Wang
TL;DR
The paper develops a multi-parameter fractional integration framework on the Heisenberg group with Zygmund dilations, introducing a fractional integral I_{a,b} via kernel V^{a,b} and a strong fractional maximal operator M_{gamma}. It establishes sharp L^p-L^q bounds under the homogeneity condition (a+b)/(n+1) = 1/p - 1/q, with the best exponent r = |a - n b|/(n+1) for I_{a,b}, and derives L^p-L^q bounds for M_{gamma} through a Córdoba-Fefferman covering lemma. A key link M_{alpha,beta} <= M_{gamma} with gamma = (alpha+beta)/(n+1) is shown, and the covering lemma underpins the weak-type and interpolation arguments. Overall, the results extend Folland-Stein type fractional inequalities to a multi-parameter setting on the Heisenberg group and provide sharp, structured bounds for both integral and maximal operators under Zygmund-type dilations.
Abstract
We study strong fractional maximal operator and fractional integral operator associated with Zygmund dilation defined on Heisenberg group. Characterizations are established for the L^p to L^q regularity of these two operators.