Abstract:

Let L be the infinitesimal generator of an analytic semigroup on L²(Rⁿ) with
Gaussian kernel bounds, and L^-α/2 be the fractional integrals generated by L for 0 < α< n. Let T_j,1 be the singular integral with nonsmooth kernel related to L, or T_j,₁ = I, T_j,₂, T_j,4 be the linear operators, which are bounded on L^p(Rⁿ) for 1 < p < 1, and T_j,3 = ±I(j = 1, 2, · · · , m), where I is the identity operator. For b ∈ L¹_loc(Rⁿ), denote the Toeplitz-type operator by

Θ^b_a f = ∑^m _j=1 (T_j,1M_bI_α T_j,₂ + T_j,3M_bI_{α Tj,4}),
where M_b is a multiplication operator. When b ∈Λ_β(0 < < 1), the authors consider the boundedness of Θ^b_α.

Key words: Toeplitz type operator, generalized fractional integral, singular integral with
nonsmooth kernel, Lipschitz function space