Behavior of bounds of singular integrals for large dimension

In this paper, we give estimates of the L^p(Rⁿ) (1 < p < ∞) and weak type (1, 1) norm of singular integral operator Tω with homogeneous kernel defined by Tωf(x) = p.v. ∫ Rⁿ ω (x - y)/|x - y|ⁿ f(y) dy, where ω satisfies L¹-Dini condition with mean value zero on S^n-1.More precisely, we show that the following two estimates hold: ||Tωf||_p ≲ ((log n)∥ω∥₁ + ^1\n ∫ ω ₁(δ)/δ dδ)∥f∥_p, and for any λ > ₀, λm({x ∈ Rⁿ : |Tωf(x)| > λ}) ≲ ((log n)∥1 + ^1/n ∫ ₀ ω₁(δ)/ δ dδ)∥f∥₁, where ω₁ denotes the L¹ integral continuousmodulus of defined by translation inRⁿ.Moreover, some similar estimates are also established for the Marcinkiewicz integral μω, which was introduced by Stein in [13].