In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{\varphi }$, when $\varphi $ is a linear-fractional self-map of $\mathbb {D}$. In this paper first, we investigate the essential normality problem for the operator $T_{w}C_{\varphi }$ on the Hardy space $H^{2}$, where $w$ is a bounded measurable function on $\partial \mathbb {D}$ which is continuous at each point of $F(\varphi )$, $\varphi \in {\cal S}(2)$, and $T_{w}$ is the Toeplitz operator with symbol $w$. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on $H^{2}$.
In this paper, it is proved that the Fourier integral operators of order $m$, with $-n < m \leq -(n-1)/2$, are bounded from three kinds of Hardy spaces associated with Herz spaces to their corresponding Herz spaces.