Toeplitz-type operators on lebesgue spaces

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 Tj,1 be the singular integral with nonsmooth kernel related to L, or Tj,1 = I, Tj,2, Tj,4 be the linear operators, which are bounded on Lp(ℝn) for 1 < p < ∞, and Tj,3 = ±I (j = 1, 2, ċ, m), where I   is the identity operator. For Lloc1 (ℝn), denote the Toeplitz-type operator by Θαbf=∑j=1m(Tj,1MbIαTj,2+Tj,3MbIαTj,4),where Mb   is a multiplication operator. When b∈Λ˙β(0<β<1), the authors consider the boundedness of Θαb.