Abstract
In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. We prove the boundedness of regular symbols on Sobolev spaces H2α(Rd;L2(M)) and Besov spaces Bp,qα(Rd;Lp(M)) for α∈R and 1≤p,q≤∞, as well as the boundedness of forbidden symbols on H2α(Rd;L2(M)) and Bp,qα(Rd;Lp(M)) for α>0 and 1≤p,q≤∞. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces F1α,c(Rd,M) obtained in our previous paper, we establish the F1α,c-regularity of regular symbols for every α∈R, and the F1α,c-regularity of forbidden symbols for α>0. As applications, we obtain the same results on the usual and quantum tori.
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