Abstract
In this note we give sufficient conditions for the L p boundedness of periodic Fourier integral operators. We also refer to them as Fourier series operators (FSOs). The main tool will be the notion of full symbol and the periodic analysis on the torus introduced by Ruzhansky and Turunen [34].
Highlights
In this note we investigate the Lp -boundedness of periodic Fourier integral operators which, by following Ruzhansky and Turunen [34], will be called Fourier series operators
If a : Tn × Zn → C, is a function defined on the phase space Tn ×Zn, where T ∼= R/Z is the one-dimensional torus, the Fourier series operator (FSO) associated with a is defined by: A f (x) := eiφ(x,ξ)a(x, ξ)(FTn f )(ξ), f ∈ C ∞(Tn ), (1)
The fundamental goal of this paper is to announce our results in [12] about the Lp -boundedness of Fourier series operators, where we cover a suitable class of non-degenerate phase functions φ on Tn × Rn, and a family of symbols a on Tn × Zn belonging to the toroidal Hörmander classes
Summary
In this note we investigate the Lp -boundedness of periodic Fourier integral operators which, by following Ruzhansky and Turunen [34], will be called Fourier series operators. Ξ∈Zn where FTn f is the toroidal Fourier transform of f These operators appear as solutions of several periodic hyperbolic problems, and their microlocal properties, in particular, an associated symbolic calculus for them, as well as, their boundedness properties on L2, were studied by. The fundamental goal of this paper is to announce our results in [12] about the Lp -boundedness of Fourier series operators, where we cover a suitable class of non-degenerate phase functions φ on Tn × Rn, and a family of symbols a on Tn × Zn belonging to the toroidal Hörmander classes (see [34, Chapter 4] for details).
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