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The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.

The α-modulation spaces$M$^{$s$,α}_{$p$,$q$}($R$^{$d$}), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ($x$,$D$) with symbol in the Hörmander class$S$^{$b$}_{ρ,0}extends to a bounded operator σ($x$,$D$):$M$^{$s$,α}_{$p$,$q$}($R$^{$d$})→$M$^{$s$-$b$,α}_{$p$,$q$}($R$^{$d$}) provided 0≤α≤ρ≤1, and 1<$p$,$q$<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class$S$^{$b$}_{1,0}maps the Besov space$B$^{$s$}_{$p$,$q$}($R$^{$d$}) into$B$^{$s$-$b$}_{$p$,$q$}($R$^{$d$}).