Abstract
A necessary and sufficient condition is imposed on the symbols σ: ℤ×S 1→ℂ to guarantee that the corresponding pseudo-differential operators T σ: L 2(ℤ)→L 2(ℤ) are Hilbert-Schmidt. A special sufficient condition on the symbols σ: ℤ×S 1→ℂ for the corresponding pseudo-differential operators T σ: L 2(ℤ)→L 2(ℤ) to be bounded is given. Sufficient conditions are given on the symbols σ: ℤ×S 1→ℂ to ensure the boundedness and compactness of the corresponding pseudo-differential operators T σ: L p(ℤ)→L p(ℤ) for 1≨p<∞. Norm estimates for the pseudo-differential operators T σ are given in terms of the symbols σ. The almost diagonalization of the pseudo-differential operators is then shown to follow from the sufficient condition for the L p-boundedness.
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Molahajloo, S. (2009). Pseudo-Differential Operators on ℤ. In: Schulze, BW., Wong, M.W. (eds) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations. Operator Theory: Advances and Applications, vol 205. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0198-6_12
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DOI: https://doi.org/10.1007/978-3-0346-0198-6_12
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