Toeplitz operators related to strongly singular Calderón-Zygmund operators

Abstract

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε $\dot \Lambda _{\beta _0 }$ (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), $\tfrac{1}{q} = \tfrac{1}{p} - \tfrac{{\beta _0 }}{n}$, and from L ^p(ℝⁿ) to Triebel-Lizorkin space $\dot F_p^{\beta _0 ,\infty }$. We also obtain the boundedness of generalized Toeplitz operator Θ ^b _α0 from L ^p(ℝⁿ) to L ^q(ℝⁿ), $\tfrac{1}{q} = \tfrac{1}{p} - \tfrac{{\alpha _0 + \beta _0 }}{n}$. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.