Microglobal regularity and the global wavefront set

Hoepfner, G.; Raich, Andrew

doi:10.1007/s00209-018-2176-0

Cited by 4 publications

(5 citation statements)

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“…For concreteness, we are going to concentrate on the cases when X is defined by a finite collection of semi-norms (e.g., Examples 2.1 and 2.2), a global L q Gevrey space [AHR17] (e.g., Example 2.3), or a global L q Denjoy-Carleman space as in [HR19] (see (3.1) below).…”

Section: General Function Classesmentioning

confidence: 99%

“…We will not need to know the requirements on M j apart from that it is increasing and satisfies (2.3) (see [HR19]). The most important example of the global L q Denjoy-Carleman spaces are the global L q -Gevrey spaces G q,s (R d ) in which case M j = j js [AHR17, HR].…”

Section: Distributions With Decaymentioning

confidence: 99%

“…First, though, we recall the FBI transform and FBI E q,M -microglobal regularity [HR19] but need to introduce one piece of notation. For ξ ∈ R d , define…”

Section: Distributions With Decaymentioning

confidence: 99%

“…Our hope was to use these function classes to explore global properties of partial differential operators, but we discovered that the Fourier transforms of such functions can be highly nonsmooth (e.g., a measure supported on a Salem set). To overcome this difficulty, we substituted the FBI transform for the FBI transform and developed microglobal tools to analyze operators [HR19]. While this program was successful for certain classes of problem, many objects are naturally analyzed with Fourier transforms, so in this paper, we develop a microglobal analysis based on a theory of distributions with decay.…”

Section: Introductionmentioning

confidence: 99%

“…(2) The (Fourier) microglobal wavefront set. In [HR19], we defined a notion of the wavefront set based on a global FBI transform. The FBI transform is difficult to compute and for objects naturally defined as tempered distributions, the Fourier transform is more natural.…”

Section: Introductionmentioning

confidence: 99%

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Distributions with Decay and Restriction Problems

Hoepfner,

Raich

2019

Preprint

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In this paper we introduce a new type of restriction problem, called the restriction problem with moments. We show that the surface area measure of the sphere satisfies the L p -L 2 restriction problem with moments if 1 ≤ p < 2(d+2) d+3 and that the Frostman measure constructed by Salem satisfies the L p -L 2 restriction problem with moments if4(1−α)+β for certain values of α and β. The main tool to obtain these new type of restriction phenomenon is the notion of distributions with decay in connection with classes of global L q ultradifferentiable functions. We develop the notion of distributions with decay and use it to define global wavefront sets of classes of function spaces, including L p -Sobolev spaces on R d as well as global L q -Denjoy Carleman functions. We also introduce the corresponding notion of microglobal regularity. We prove a characterization of distributions (in a given function space) with decay in terms of microglobal regularity in every direction of their Fourier transforms.

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Section: General Function Classesmentioning

confidence: 99%

Section: Distributions With Decaymentioning

confidence: 99%

“…First, though, we recall the FBI transform and FBI E q,M -microglobal regularity [HR19] but need to introduce one piece of notation. For ξ ∈ R d , define…”

Section: Distributions With Decaymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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