L^p 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 ∈ L¹(Σ) and satisfies the cancellation condition of order [α], then for any 1 < p < ∞,
∥T^#_Ω,αf∥L^p(ℍ) ≲ ∥Ω∥L¹(Σ)∥f∥L^p_α(ℍ).
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 ∈ Ap,
∥T^#_Ω,αf∥L^p(w) ≲ ∥Ω∥L^q(Σ) {w} A_p (w) A_p ∥f∥L^p_α(w), 1< p < ∞.