Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

作者：Karlovich Yuri I^*; Loreto Hernandez Ivan

来源：Integral Equations and Operator Theory, 2012, 74(3): 377-415.

DOI：10.1007/s00020-012-2004-6

• Convolution type operator
• piecewise slowly oscillating function
• commutator
• maximal ideal space
• local principle
• limit operator
• two idempotents theorem

Let be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space L-p(R,w) where p is an element of (1, infinity) and w is a Muckenhoupt weight. We study the Banach subalgebra u(p,w) of B-p,B-w generated by all multiplication operators aI (a is an element of PSO lozenge) and all convolution operators W-0(b) (b is an element of PSOp,w lozenge), where PSO lozenge subset of L-infinity(R) and PSOp,w lozenge subset of M-p,M-w are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of RU{infinity}, and M (p,w) is the Banach algebra of Fourier multipliers on L-p(R, w). Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra u(p,w) and establish a Fredholm criterion for the operators A is an element of u(p,w) in terms of their Fredholm symbols. To study the Banach algebra u(p,w) we apply the theory of Mellin pseudodifferential operators, the Allan-Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of u(p,w) and necessary tools for studying local algebras.

• 出版日期2012-11