Boundedness of Singular Integral Operators on Local Hardy Spaces and Dual Spaces

Abstract

The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the singular integrals considered in this paper include the pseudo-differential operators $T_{\sigma }f(x)=\int \limits \sigma (x \xi )e^{2\pi ix\xi }\hat {f}(\xi )d\xi $ with $\sigma \in S_{1 0}^{0}$ . As a consequence our results give another proof of the boundedness of the pseudo-differential operators on the local Hardy space (Goldberg Duke Math. J. 46(1) 27–42 1979).