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Pseudodifferential operators of Mellin type. (English) Zbl 0532.35085
Pseudodifferential operators of Mellin type on the half line \({\mathbb{R}}_+\) are defined by the authors by means of the formula \[ a(t,- td/dt)f(t)=(2\pi i)^{-1}\int_{Re z=1/p}t^{-z}a(t,z)\tilde f(z)dz, \] where the symbol a(t,z) satisfies suitable smoothness properties in t and holomorphy and growth properties in z, and \(\tilde f\)(z)\(=\int^{+\infty}_{0}t^{z-1}f(t)dt\) is the Mellin transform of \(f\in L^ p({\mathbb{R}}_+)\). For operators of this type the authors develop a detailed symbolic calculus, proving results of continuity on \(L^ p\) spaces, weighted \(L^ p\) spaces and related Sobolev spaces. A condition of ellipticity is introduced, under which the operators are Fredholm; the index of the Fredholm operators is also computed. Similar operators are also considered in a finite interval I in \({\mathbb{R}}.\)
Such a calculus finds natural application in the study of Dirichlet, oblique derivative and mixed boundary value problems in a plane polygon. In this connection, the authors recapture and generalize several known results, as it is typical for these problems, Fredholm conditions and index formulas depend on the \(L^ p\) space on which the problem is set.
Reviewer: L.Rodino

35S05 Pseudodifferential operators as generalizations of partial differential operators
47Gxx Integral, integro-differential, and pseudodifferential operators
47A53 (Semi-) Fredholm operators; index theories
47A60 Functional calculus for linear operators
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