Two points, one limit: Homogenization techniques for two-point local algebras
Copyright policy )Two points, one limit: Homogenization techniques for two-point local algebras
This paper studies several algebras generated by convolution, multiplication and flip operators on \(L^p(\mathbb R)\). Let \(\mathcal{A}\) be the smallest closed subalgebra of \(\mathcal{L}(L^p(\mathbb R))\) which contains multiplication operators \(aI\), Fourier convolution operators \(W^0(b)\) with \(a\) and \(b\) piecewise continuous, and the flip operator \(J\). The main result can be stated as follows. Theorem. There is a family of algebra homomorphisms \(Y_{s,t}\) labeled by the points in \(([0,\infty]X\{\infty\})\times (\{\infty\}X[0,\infty])\) such that an operator \(a\in \mathcal{A}\) is Fredholm on \(L^p(\mathbb R)\) if and only if all operators \(Y_{s,t}(A)\) are invertible. The result explores the properties of the Fourier transform in \(L^p(\mathbb R)\), when \(p\neq 2\).
- TU Darmstadt Germany
- Technische Universität Darmstadt Germany
- Technische Universität Darmstradt Germany
- Technical University of Lisbon Portugal
- UNIVERSIDADE TECNICA DE LISBOA - UTL Portugal
Structure theory of linear operators, Homogenization techniques, Applied Mathematics, Convolution operators, convolution operators, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.), Convolution as an integral transform, local principles, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Classical operational calculus, homogenization techniques, Banach algebras, Local principles, Analysis
Structure theory of linear operators, Homogenization techniques, Applied Mathematics, Convolution operators, convolution operators, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.), Convolution as an integral transform, local principles, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Classical operational calculus, homogenization techniques, Banach algebras, Local principles, Analysis
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