Ultradifferentiable functions and Fourier analysis

Braun, Rüdiger W.; Meise, Reinhold; Taylor, B. A.

doi:10.1007/bf03322459

Cited by 281 publications

(494 citation statements)

References 12 publications

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“…The topological structure of these spaces is much more involved than that of Fréchet spaces of C ∞ functions. The main result is based on Theorem 2.1 about the existence of local two-sided kernels, which extends (and improves) the work of Morando [20] from Gevrey classes to the case of general non-quasianalytic classes as defined by Braun, Meise and Taylor [3]; see the details below. Our results are obtained as an application of topological tensor products and spaces of vector valued ultradistributions and ultradifferentiable functions to linear partial differential operators.…”

Section: §1 Introduction and Preliminariesmentioning

confidence: 57%

“…If F is complete, then we have the following canonical isomorphism The theory of kernels of L. Schwartz can be extended to the setting of the classes of {ω}-ultradifferentiable functions and {ω}-ultradistributions. Indeed, using well known results of Grothendieck [6] on topological tensor products one can prove the following results (see, e.g., [12,13,3]). …”

Section: Definition 12mentioning

confidence: 93%

“…which is the strong dual of a nuclear Fréchet space (i.e., a (DFN)-space) if it is endowed with its natural inductive limit topology [3].…”

Section: Definition 12mentioning

confidence: 99%

“…We fix notation and give some definitions and results which will be useful for the sequel. Following Braun, Meise and Taylor [3], we introduce the classes of non-quasianalytic functions of Roumieu type. …”

Section: §1 Introduction and Preliminariesmentioning

confidence: 99%

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Ultradifferentiable Fundamental Kernels of Linear Partial Diﬀerential Operators on Non-quasianalytic Classes of Roumieu Type

Albanese

Bonet

2007

Publ. Res. Inst. Math. Sci.

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Let P be a linear partial differential operator with coefficients in the Roumieu class E {ω} (Ω). We prove that if P and its transposed operator t P are {ω}-hypoelliptic in Ω and surjective on the space E {ω} (Ω), then P has a global two-sided ultradifferentiable fundamental kernel in Ω, thus extending to the Roumieu classes the well-known analogous result of B. Malgrange in the C ∞ class. This result is new even for Gevrey classes. §1. Introduction and PreliminariesEhrenpreis [5] and Malgrange [17] proved that every linear partial differential operator with constant coefficients has a fundamental solution; therefore the inhomogeneous equation P u = f admits always C ∞ /E {ω} -solutions for each right hand side f in the class C ∞ /E {ω} respectively, and with compact support. However, it is well-known that this is not longer the case for linear partial differential operators with variable coefficients, i.e., see the Lewy's operator. The

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Section: §1 Introduction and Preliminariesmentioning

confidence: 57%

Section: Definition 12mentioning

confidence: 93%

“…which is the strong dual of a nuclear Fréchet space (i.e., a (DFN)-space) if it is endowed with its natural inductive limit topology [3].…”

Section: Definition 12mentioning

confidence: 99%

Section: §1 Introduction and Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Ultradifferentiable Fundamental Kernels of Linear Partial Diﬀerential Operators on Non-quasianalytic Classes of Roumieu Type

Albanese

Bonet

2007

Publ. Res. Inst. Math. Sci.

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“…Motivated by the recent work developed in [15,16,32,33,43] and in [2,3], we investigate the global hypoellipticity of linear partial differential operators defined on the torus T N in a bigger scale of spaces, namely, in the setting of ultradifferentiable classes as introduced in [10]. Actually, we prove the ω-regularity of solutions of operators of type P = P (t, D t , Dx) defined on the torus T m+n with real valued coefficients in E * (T m ) and which are globally hypoelliptic in T m+n .…”

Section: Introductionmentioning

confidence: 99%