(untitled) (Analysis in Theory and Applications, 2026)
In this article we consider a modification of the Stein’s spherical maximal operator of complex order α on R^n: M_[1,2]^α f(x) = _t [1,2] | 1Γ(α) _|y| ≤ 1 ( 1 - |y|^2 )^α - 1 f(x - ty) dy |. We show that when n ≥ 2, suppose \|M_[1,2]^α f\|_L^q(R^n) ≤ C \|f\|_L^p(R^n) holds for some α C, p, q ≥ 1, then we must have that q ≥ p and Re α ≥ σ_n(p, q) := { 1p - nq, n+12p - n-12 ( 1q + 1 ), np - n + 1 }. Conversely, we show that M_[1,2]^α is bounded from L^p(R^n) to L^q(R^n) provided that q ≥ p and Re α > σ_2(p, q) for n = 2; and Re α > { σ_n(p, q), 1/(2p) - (n-2)/(2q) - (n-1)/4 } for n > 2. The range of α, p and q is almost optimal in the case when either n = 2, or α = 0, or (p, q) lies in certain regions for n > 2.