On a Problem of Pichorides
(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.
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- author
- Bakas, Odysseas LU
- organization
- publishing date
- 2021-07-01
- 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 < p< 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<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 <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}}, }