Premium
An Algebra of Pseudodifferential Operators with Slowly Oscillating Symbols
Author(s) -
Karlovich Yu. I.
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1017/s0024611505015674
Subject(s) - mathematics , banach algebra , quotient , bounded function , banach space , pure mathematics , differentiable function , pseudodifferential operators , algebra over a field , mathematical analysis
Let V R denote the Banach algebra of absolutely continuous functions of bounded total variation on R, and let B p be the Banach algebra of bounded linear operators acting on the Lebesgue space L p R for 1 < p < ∞. We study the Banach algebra A ⊂ B p generated by the pseudodifferential operators of zero order with slowly oscillating V R‐valued symbols on R. Boundedness and compactness conditions for pseudodifferential operators with symbols in L ∞ R, V R are obtained. A symbol calculus for the non‐closed algebra of pseudodifferential operators with slowly oscillating V R‐valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. As a result, the quotient Banach algebra A π = A K, where K is the ideal of compact operators in B p , is commutative and involutive. An isomorphism between the quotient Banach algebra A π of pseudodifferential operators and the Banach algebraA ^of their Fredholm symbols is established. A Fredholm criterion and an index formula for the pseudodifferential operators A ∈ A are obtained in terms of their Fredholm symbols. 2000 Mathematics Subject Classification 47G30, 47L15 (primary), 47A53, 47G10 (secondary).