Localization properties of highly singular generalized functions

Смирнов, Александр Георгиевич

doi:10.1007/s11232-007-0046-8

Cited by 1 publication

(5 citation statements)

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“…By Lemma 16, there exist r, r ′ ≥ 0 and a plurisubharmonic function Φ such that (31)-(33) hold for K = V2 and V = (R k \ V1 ) ∪{0}. Since the space S 0 α (R) is nontrivial, Lemma 5 in [7] ensures that there are constants B > 0 and H and a plurisubharmonic function σ on C k such that…”

Section: The Case Of Proper Conesmentioning

confidence: 99%

“…The next elementary lemma, which follows from Lemma 9 in [7], summarizes some simple facts about cones in R k needed for the proof of Theorem 9. Lemma 11.…”

Section: The Case Of Proper Conesmentioning

confidence: 99%

“…Since the space S 0 α (R) is nontrivial, Lemma 5 in [7] ensures that there are constants B > 0 and H and a plurisubharmonic function σ on C k such that −|x| 1/α − H σ (x + iy) −|x| 1/α + B| y|, x, y ∈ R k .…”

Section: The Case Of Proper Conesmentioning

confidence: 99%

“…However, this strategy implies using L 2 -type norms, while the spaces S 0 α (U) are defined by supremum norms. To pass to L 2 -norms, we make use of results of [7], where this problem was considered for a broad class of spaces containing all spaces S 0 α (U) with α ≥ 1. Given α ≥ 1, A, B > 0, and a cone U in R k , let H 0,B α,A (U) be the Hilbert space of entire functions on C k with the finite norm…”

Section: The Case Of Proper Conesmentioning

confidence: 99%

“…where W ranges all conic neighborhoods of U and the union is endowed with the inductive limit topology. The next elementary lemma, which follows from Lemma 9 in [7], summarizes some simple facts about cones in R k needed for the proof of Theorem 9.…”

Section: The Case Of Proper Conesmentioning

confidence: 99%

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On localization properties of Fourier transforms of hyperfunctions

Smirnov

2009

Journal of Mathematical Analysis and Applications

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In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space $\mathcal U(R^k)$ which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown that all Gelfand--Shilov spaces $S^{\prime 0}_\alpha(R^k)$ ($\alpha>1$) of analytic functionals are canonically embedded in $\mathcal U(R^k)$. While the usual definition of support of a generalized function is inapplicable to elements of $S^{\prime 0}_\alpha(R^k)$ and $\mathcal U(R^k)$, their localization properties can be consistently described using the concept of {\it carrier cone} introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49-59; Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of $S^{\prime 0}_\alpha(R^k)$ and $\mathcal U(R^k)$ is studied. It is proved that an analytic functional $u\in S^{\prime 0}_\alpha(R^k)$ is carried by a cone $K\subset R^k$ if and only if its canonical image in $\mathcal U(R^k)$ is carried by $K$.Comment: 21 pages, final version, accepted for publication in J. Math. Anal. App

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Section: The Case Of Proper Conesmentioning

confidence: 99%

“…The next elementary lemma, which follows from Lemma 9 in [7], summarizes some simple facts about cones in R k needed for the proof of Theorem 9. Lemma 11.…”

Section: The Case Of Proper Conesmentioning

confidence: 99%

Section: The Case Of Proper Conesmentioning

confidence: 99%

Section: The Case Of Proper Conesmentioning

confidence: 99%

Section: The Case Of Proper Conesmentioning

confidence: 99%

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