Composition of some singular Fourier integral operators and estimates for restricted X -ray transforms

Allan Greenleaf; Gunther Uhlmann

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 2, page 443-466
  • ISSN: 0373-0956

Abstract

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We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations C ⊂ ( T * X ∖ 0 ) × ( T * Y ∖ 0 ) . These canonical relations, which arise naturally in integral geometry, are such that π : C → T * Y is a Whitney fold and ρ : C → T * X is a blow-down mapping. If A ∈ I m ( C ) , B ∈ I m ' ( C t ) , then B A ∈ I m + m ' , 0 ( Δ , Λ ) a class of pseudodifferential operators with singular symbols. From this follows L 2 boundedness of A with a loss of 1/4 derivative.

How to cite

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Greenleaf, Allan, and Uhlmann, Gunther. "Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms." Annales de l'institut Fourier 40.2 (1990): 443-466. <http://eudml.org/doc/74884>.

@article{Greenleaf1990,
abstract = {We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations $C\subset (T^*X\setminus 0)\times (T^*Y\setminus 0)$. These canonical relations, which arise naturally in integral geometry, are such that $\pi $ : $C\rightarrow T^*Y$ is a Whitney fold and $\rho $ : $C\rightarrow T^*X$ is a blow-down mapping. If $A\in I^ m(C)$, $B\in I^\{m^\{\prime \}\}(C^ t)$, then $BA\in I^\{m+m^\{\prime \},0\}(\Delta ,\Lambda )$ a class of pseudodifferential operators with singular symbols. From this follows $L^ 2$ boundedness of $A$ with a loss of 1/4 derivative.},
author = {Greenleaf, Allan, Uhlmann, Gunther},
journal = {Annales de l'institut Fourier},
keywords = {Fourier integral operators},
language = {eng},
number = {2},
pages = {443-466},
publisher = {Association des Annales de l'Institut Fourier},
title = {Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms},
url = {http://eudml.org/doc/74884},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Greenleaf, Allan
AU - Uhlmann, Gunther
TI - Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 2
SP - 443
EP - 466
AB - We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations $C\subset (T^*X\setminus 0)\times (T^*Y\setminus 0)$. These canonical relations, which arise naturally in integral geometry, are such that $\pi $ : $C\rightarrow T^*Y$ is a Whitney fold and $\rho $ : $C\rightarrow T^*X$ is a blow-down mapping. If $A\in I^ m(C)$, $B\in I^{m^{\prime }}(C^ t)$, then $BA\in I^{m+m^{\prime },0}(\Delta ,\Lambda )$ a class of pseudodifferential operators with singular symbols. From this follows $L^ 2$ boundedness of $A$ with a loss of 1/4 derivative.
LA - eng
KW - Fourier integral operators
UR - http://eudml.org/doc/74884
ER -

References

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