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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.

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.)

Citations:

Zbl 0741.00022
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