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).

中文翻译：

---

局部Hardy空间和对偶空间上奇异积分算子的有界性。

本文的目的是为局部Hardy空间及其对偶上的奇异积分的有界性提供必要和充分的条件。特别是本文考虑的奇异积分包括伪微分算子

$ T _ {\ sigma} f（x）= \ int \ limits \ sigma（x \ xi）e ^ {2 \ pi ix \ xi} \ hat {f}（\ xi）d \ xi $

与\（\ sigma \ in S_ {1 0} ^ {0} \）。结果，我们的结果再次证明了局部Hardy空间上伪微分算子的有界性（Goldberg Duke Math。J. 46（1）27-42 1979）。