The FBI transforms and their use in microlocal analysis

Hoepfner, G.; Medrado, R.

doi:10.1016/j.jfa.2018.05.022

Cited by 11 publications

(12 citation statements)

References 18 publications

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“…In the literature (for instance, see [1] and [10]), it is usual to state this inequality by means of the associated function as follows…”

Section: Condition C) Meansmentioning

confidence: 99%

“…">1.We can also define

\tilde{h} false(r false) ≐ {trueprefixinf}_{k \in {double-struckZ}_{+}} M_{k} r^{k}

. With this notation the inequality (2.1) takes the form

|scriptF ([], χ, u) (x, ξ)| \leq A \tilde{h} (A {| ξ |}^{- 1}) .

In the literature (for instance, see [1] and [10]), it is usual to state this inequality by means of the associated function as follows

|scriptF ([], χ, u) (x, ξ)| \leq A_{1} e^{- M (A_{2} | ξ |)},

with

A_{1}, A_{2} > 0

. 2.Notice that for

r \geq 1

, we have

h_{1} false(r false) = 1

.…”

Section: Denjoy–carleman Classesmentioning

confidence: 99%

See 1 more Smart Citation

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Rodrigues

Silva

2021

Mathematische Nachrichten

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In this paper we study microlocal regularity of a  2 -solution of the equation) is ultradifferentiable in the variables ( , ) ∈ ℝ × ℝ and holomorphic in the variables () ∈ ℂ × ℂ . We proved that if   is a regular Denjoy-Carleman class (including the quasianalytic case) then:where WF  ( ) is the Denjoy-Carleman wave-front set of and Char( ) is the characteristic set of the linearized operator :

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“…In the literature (for instance, see [1] and [10]), it is usual to state this inequality by means of the associated function as follows…”

Section: Condition C) Meansmentioning

confidence: 99%

“…">1.We can also define

\tilde{h} false(r false) ≐ {trueprefixinf}_{k \in {double-struckZ}_{+}} M_{k} r^{k}

. With this notation the inequality (2.1) takes the form

|scriptF ([], χ, u) (x, ξ)| \leq A \tilde{h} (A {| ξ |}^{- 1}) .

In the literature (for instance, see [1] and [10]), it is usual to state this inequality by means of the associated function as follows

|scriptF ([], χ, u) (x, ξ)| \leq A_{1} e^{- M (A_{2} | ξ |)},

with

A_{1}, A_{2} > 0

. 2.Notice that for

r \geq 1

, we have

h_{1} false(r false) = 1

.…”

Section: Denjoy–carleman Classesmentioning

confidence: 99%

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Rodrigues

Silva

2021

Mathematische Nachrichten

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“…Here we shall use the following characterization of Denjoy-Carleman wave front that can be found in [20]. (2.9)…”

Section: Denjoy-carleman Functionsmentioning

confidence: 99%

“…The associated function of a sequence (M j ) as above, is defined by M(t) := sup p log (t p /M p ). In the next lemma we recall an important property of the associated function, proved in [20] and [28]. where A and H were defined by (A 7).…”

Section: Appendix a Denjoy-carleman Classesmentioning

confidence: 99%

Microlocal Regularity for Mizohata Type Differential Operators

Hoepfner¹,

Medrado²

2018

J. Inst. Math. Jussieu

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“…Let S be one of these sheaves then the microlocal regularity with respect to S can be classified in terms of estimates of the FBI transform by the condition M(S), see Section 6. We avail ourselves of results in [BH12,Für18,HM18] as ingredients which allowed us to use our technique to extend them to this richer framework.…”

Section: Introductionmentioning

confidence: 99%

A new microlocal analysis of hyperfunctions

Hoepfner,

Ragognette

2021

Preprint

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In this work we study microlocal regularity of hyperfunctions defining in this context a class of generalized FBI transforms first introduced for distributions by Berhanu and Hounie. Using a microlocal decomposition of a hyperfunction and the generalized FBI transforms we were able to characterize the wave-front set of hyperfunctions according several types of regularity. The microlocal decomposition allowed us to recover and generalize both classical and recent results and, in particular, we proved for differential operators with real-analytic coefficients that if the elliptic regularity theorem regarding any reasonable regularity holds for distributions, then it is automatically true for hyperfunctions.

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The FBI transforms and their use in microlocal analysis

Cited by 11 publications

References 18 publications

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Microlocal Regularity for Mizohata Type Differential Operators

A new microlocal analysis of hyperfunctions

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