Lp Estimates and Weighted Estimates of Fractional Maximal Rough Singular Integrals on Homogeneous Groups.

In this paper, we study the L p boundedness and L p (w) boundedness ( 1 < p < ∞ and w a Muckenhoupt A p weight) of fractional maximal singular integral operators T Ω , α # with homogeneous convolution kernel Ω (x) on an arbitrary homogeneous group H of dimension Q . We show that if 0 < α < Q , Ω ∈ L 1 (Σ) and satisfies the cancellation condition of order [ α ] , then for any 1 < p < ∞ , ‖ T Ω , α # f ‖ L p (H) ≲ ‖ Ω ‖ L 1 (Σ) ‖ f ‖ L α p (H) . <graphic href="12220_2022_1007_Article_Equ146.gif"></graphic> where for the case α = 0 , the L p boundedness of rough singular integral operator and its maximal operator were studied by Tao (Indiana Univ Math J 48:1547–1584, 1999) and Sato (J Math Anal Appl 400:311–330, 2013), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if 0 ≤ α < Q and Ω satisfies the same cancellation condition but a stronger condition that Ω ∈ L q (Σ) for some q > Q / α , then for any 1 < p < ∞ and w ∈ A p , ‖ T Ω , α # f ‖ L p (w) ≲ ‖ Ω ‖ L q (Σ) { w } A p (w) A p ‖ f ‖ L α p (w) , 1 < p < ∞. <graphic href="12220_2022_1007_Article_Equ147.gif"></graphic> [ABSTRACT FROM AUTHOR]