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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

##### MSC:
 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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