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Pseudo-differential and Mellin operators in spaces with conormal singularity (boundary symbols). (English) Zbl 0542.35001

Rep., Akad. Wiss. DDR, Inst. Math. R-MATH-01/84, 96 p. (1984).
Authors’ summary: The paper develops the boundary symbolic calculus belonging to boundary value problems for pseudo-differential operators (\(\psi\) DO’s) without the transmission property. First there are studied function spaces \({\mathcal S}_{{\mathfrak p}}(C^{\infty}_{{\mathfrak p,p'}})\) on \({\mathbb{R}}_+\) which are in \(C^{\infty}({\mathbb{R}}_+)\) and have a so- called conormal asymptotics in t (at 0 and \(\infty\), respectively) of type \({\mathfrak p}({\mathfrak p,p'})\). The boundary symbols are (modulo a reducing to left upper corner) parameter depending operators of the form \[ (1)\quad {\mathfrak a}(\xi ')=op_+(a)(\xi ')+\sum^{\infty}_{j=0}t^{-j}\omega(c_ jt| \xi '|)op_ M(h_ j)(\xi ')\omega(t| \xi '|)+op_ G(b)(\xi '), \] where \(\xi '\in {\mathbb{R}}^{n-1}\backslash 0\) is the parameter tangent to the boundary belonging to the half-space theory, and \(op_+(a)\) is a \(\psi\) DO on \({\mathbb{R}}_+\), \(op_ M(h_ j)\) is a Mellin operator with the symbol \(h_ j\), \(c_ j\in {\mathbb{R}}\) constants, \(c_ j\to +\infty\), \(\omega(t)\in C^{\infty}({\bar {\mathbb{R}}}_+)\) a cut-off function, and \(op_ G(b)\) a Green operator. The symbols involved in \({\mathfrak a}(\xi ')\) are connected with the spaces \({\mathcal S}_{{\mathfrak p}}\) and \(C^{\infty}_{{\mathfrak p,p'}}\), respectively, with certain \({\mathfrak p,p'}\), e.g. \(b(\xi ',t,s)\in C^{\infty}({\mathbb{R}}_{\xi '}^{n- 1}\backslash 0,{\mathcal S}_{{\mathfrak q}}\otimes {\mathcal S}_{{\mathfrak x}})\) with certain singularity types \({\mathfrak q,x}\). The operators (1) form an algebra with a complete symbolic calculus, and under some ellipticity condition for the interior symbol \(a=\sigma_{\psi}({\mathfrak a})\) and the principal conormal symbol \(\sigma^ 0_ N({\mathfrak a})\) we have a parametrix of a similar type (i.e. with a certain modification of the lower order Mellin terms) belonging to the inverse symbols. The operators act between functions of the type \({\mathcal S}_{{\mathfrak p}}\) in a reasonable way, and \({\mathcal S}_{{\mathfrak p}}\) being the inductive limit of all \({\mathcal S}_{{\mathfrak p}}\), \({\mathfrak p}\) any singularity type, is a natural substitute of the space of functions which are \(C^{\infty}\) up to the boundary occurring in the case of the transmission property.
A basic feature of the theory is that besides the complete interior symbols \(a(\xi)\sim \sum a_{-j}\) in \(S_{cl}\) we also have a sequence of conormal symbols \(\sigma_ N^{-j}({\mathfrak a})\) connected with the \(h_ j\) in (1) and a. Another point is that the Green operators, which also carry singularity, occur in commutation relations between Mellin operators and powers of t as well as in compositions, and they are negligible in the sense of the symbolic calculus. For applying the theory we prove the theorem that, if \({\mathfrak a}(\xi ')\) is of the type (1) and invertible in \(L^ 2({\mathbb{R}}_+)\), the inverse \({\mathfrak a}^{-1}(\xi ')\) again belongs to the algebra. The theory also applies in the case of interior symbols a that depend on \(x=(x',t)\in {\mathbb{R}}^{n-1}\times {\mathbb{R}}\). Moreover, it is valid also for systems.
Reviewer: J.Wloka

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35S15 Boundary value problems for PDEs with pseudodifferential operators
47Gxx Integral, integro-differential, and pseudodifferential operators