Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

On a Problem of Pichorides

Bakas, Odysseas LU (2021) In Journal of Geometric Analysis 31(7). p.7455-7512
Abstract

Let S(Λ) denote the classical Littlewood–Paley operator formed with respect to a lacunary sequence Λ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of S(Λ) from the analytic Hardy space HAp(T) to Lp(T) and of the behaviour of the Lp(T) → Lp(T) operator norm of S(Λ) (1 < p< 2) in terms of the ratio of the lacunary sequence Λ. Namely, if ρΛ denotes the ratio of Λ , then we prove that sup‖f‖Lp(T)=1f∈HAp(T)‖S(Λ)(f)‖Lp(T)≲1p-1(ρΛ-1)-1/2(1<p<2)and ‖S(Λ)‖Lp(T)→Lp(T)≲1(p-1)3/2(ρΛ-1)-1/2(1<p<2)and that these results are optimal as p→ 1 +. Variants in... (More)

Let S(Λ) denote the classical Littlewood–Paley operator formed with respect to a lacunary sequence Λ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of S(Λ) from the analytic Hardy space HAp(T) to Lp(T) and of the behaviour of the Lp(T) → Lp(T) operator norm of S(Λ) (1 < p< 2) in terms of the ratio of the lacunary sequence Λ. Namely, if ρΛ denotes the ratio of Λ , then we prove that sup‖f‖Lp(T)=1f∈HAp(T)‖S(Λ)(f)‖Lp(T)≲1p-1(ρΛ-1)-1/2(1<p<2)and ‖S(Λ)‖Lp(T)→Lp(T)≲1(p-1)3/2(ρΛ-1)-1/2(1<p<2)and that these results are optimal as p→ 1 +. Variants in higher dimensions and in the Euclidean setting are also obtained.

(Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Hardy spaces, lacunary sequences, Littlewood–Paley square function, Orlicz spaces
in
Journal of Geometric Analysis
volume
31
issue
7
pages
58 pages
publisher
Springer
external identifiers
  • scopus:85096026385
ISSN
1050-6926
DOI
10.1007/s12220-020-00550-8
language
English
LU publication?
yes
id
43729bae-d94e-4bda-a8c3-8307e74016f6
date added to LUP
2020-11-25 07:30:35
date last changed
2022-04-26 22:05:15
@article{43729bae-d94e-4bda-a8c3-8307e74016f6,
  abstract     = {{<p>Let S<sup>(Λ)</sup> denote the classical Littlewood–Paley operator formed with respect to a lacunary sequence Λ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of S<sup>(Λ)</sup> from the analytic Hardy space HAp(T) to L<sup>p</sup>(T) and of the behaviour of the L<sup>p</sup>(T) → L<sup>p</sup>(T) operator norm of S<sup>(Λ)</sup> (1 &lt; p&lt; 2) in terms of the ratio of the lacunary sequence Λ. Namely, if ρ<sub>Λ</sub> denotes the ratio of Λ , then we prove that sup‖f‖Lp(T)=1f∈HAp(T)‖S(Λ)(f)‖Lp(T)≲1p-1(ρΛ-1)-1/2(1&lt;p&lt;2)and ‖S(Λ)‖Lp(T)→Lp(T)≲1(p-1)3/2(ρΛ-1)-1/2(1&lt;p&lt;2)and that these results are optimal as p→ 1 <sup>+</sup>. Variants in higher dimensions and in the Euclidean setting are also obtained.</p>}},
  author       = {{Bakas, Odysseas}},
  issn         = {{1050-6926}},
  keywords     = {{Hardy spaces; lacunary sequences; Littlewood–Paley square function; Orlicz spaces}},
  language     = {{eng}},
  month        = {{07}},
  number       = {{7}},
  pages        = {{7455--7512}},
  publisher    = {{Springer}},
  series       = {{Journal of Geometric Analysis}},
  title        = {{On a Problem of Pichorides}},
  url          = {{http://dx.doi.org/10.1007/s12220-020-00550-8}},
  doi          = {{10.1007/s12220-020-00550-8}},
  volume       = {{31}},
  year         = {{2021}},
}