A note on singular integral

Abstract

Abstract In this paper, we prove that for n 2 - 1 2 < α < n 2 {\frac{n}{2}-\frac{1}{2}<\alpha<\frac{n}{2}} , α ∉ ℕ {\alpha\notin\mathbb{N}} , the convolution operator S α ⁢ f ⁢ ( x ) = ∫ | y | ≥ 1 f ⁢ ( x - y ) ⁢ ( | y | 2 - 1 ) - α ⁢ 𝑑 y S_{\alpha}f(x)=\int_{|y|\geq 1}f(x-y)(|y|^{2}-1)^{-\alpha}\,dy is bounded from L p {L^{p}} to L q 1 + L q 2 {L^{q_{1}}+L^{q_{2}}} for certain values of p and q 1 , q 2 {q_{1},q_{2}} .