Characterizations of Hardy spaces for Fourier integral operators

Rozendaal, Jan

doi:10.4171/rmi/1246

Cited by 10 publications

(6 citation statements)

References 27 publications

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“…In [16] and [7] these spaces were defined in terms of conical square functions, or equivalently tent spaces, over the cosphere bundle. That H p F IO (R n ) can be equivalently described in this manner was shown by the author in [13] for 1 < p < ∞, and together with Fan, Liu and Song in [4] for p = 1.…”

Section: Hardy Spaces For Fourier Integral Operatorsmentioning

confidence: 73%

“…However, the resulting interval would be substantially smaller than that in Theorem 5.1. Moreover, the proof of Lemma 4.3 in [14], which uses atomic decompositions of tent spaces, and then relates to the present article through an equivalent characterization of H p F IO (R n ) from [13], is not significantly simpler than that of Proposition 4.7.…”

Section: Resultsmentioning

confidence: 98%

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Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II

Rozendaal

2021

Preprint

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We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols a(x, η) are elements of C r * S m 1,δ classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces H p,s F IO (R n ) and H p,t F IO (R n ) over the Hardy space for Fourier integral operators H p F IO (R n ). Our main result is that for all r > 0, m = 0 and δ = 1/2, there exists an interval of p around 2 such that a(x, D) acts boundedly on H p F IO (R n ).

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Section: Hardy Spaces For Fourier Integral Operatorsmentioning

confidence: 73%

Section: Resultsmentioning

confidence: 98%

Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II

Rozendaal

2021

Preprint

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“…In [17] and [8] they were defined in terms of conical square functions, or equivalently tent spaces, over the cosphere bundle. That H p F I O (R n ) can be equivalently described in this manner was shown by the author [15] for 1 < p < ∞, and by Fan, Liu, Song and the author [4] for p = 1.…”

Section: Andmentioning

confidence: 78%

“…However, the resulting interval would be substantially smaller than that in Theorem 5.1. Moreover, the proof in [14] of Lemma 4.3, which relies on [8,Theorem 6.10] and on an equivalent characterization of [15], is no simpler than that of Proposition 4.7.…”

Section: Remark 53mentioning

confidence: 99%

Rough Pseudodifferential Operators on Hardy Spaces for Fourier Integral Operators II

Rozendaal

2022

J Fourier Anal Appl

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We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $$a(x,\eta )$$ a ( x , η ) are elements of $$C^{r}_{*}S^{m}_{1,\delta }$$ C ∗ r S 1 , δ m classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces $${\mathcal {H}}^{s,p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO s , p ( R n ) and $${\mathcal {H}}^{t,p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO t , p ( R n ) over the Hardy space for Fourier integral operators $${\mathcal {H}}^{p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO p ( R n ) . Our main result is that for all $$r>0$$ r > 0 , $$m=0$$ m = 0 and $$\delta =1/2$$ δ = 1 / 2 , there exists an interval of p around 2 such that a(x, D) acts boundedly on $${\mathcal {H}}^{p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO p ( R n ) .

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“…Moreover, one could solve nonlinear wave equations with initial data in H s,p F IO (R d ) in the same manner as we do for B s p,2,2 (R d ). Our goal in this article is not to develop a full theory of Besov spaces adapted to the half-wave group, as has been done for the Hardy spaces for FIOs in [8,10,21]. The advantage of working with Besov spaces is that it suffices to obtain estimates on dyadic frequency annuli, instead of working with square functions.…”

Section: Introductionmentioning

confidence: 99%