Birman, M. Sh.; Karadzhov, G. E.; Solomyak, M. Z. Boundedness conditions and spectrum estimates for the operators \(b (X) a (D)\) and their analogs. (English) Zbl 0768.47025 Estimates and asymptotics for discrete spectra of integral and differential equations, Pap. Semin. Math. Phys., Leningrad/Russia 1989-90, Adv. Sov. Math. 7, 85-106 (1991). Summary: [For the entire collection see Zbl 0741.00022.]Some conditions on functions \(a\), \(b\) on \(\mathbb{R}^ d\) are found which guarantee the boundedness of the operator \(b(X)a(D)\) on \(L_ p(\mathbb{R}^ d)\). Connections with general Hardy-type inequalities are discussed. For \(p=2\), some estimates of the \(s\)-numbers of \(b(X)a(D)\) are considered as well. In fact, the main results of the paper are related to a certain class of operators containing \(b(X)a(D)\) as a particular case. Cited in 28 Documents MSC: 47G30 Pseudodifferential operators 35S30 Fourier integral operators applied to PDEs 47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) Keywords:boundedness; Hardy-type inequalities; \(s\)-numbers Citations:Zbl 0741.00022 PDFBibTeX XMLCite \textit{M. Sh. Birman} et al., Adv. Sov. Math. 7, 85--106 (1991; Zbl 0768.47025)