A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10456382" target="_blank" >RIV/00216208:11320/22:10456382 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=yq9Tr3N_5T" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=yq9Tr3N_5T</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jfa.2021.109295" target="_blank" >10.1016/j.jfa.2021.109295</a>
Alternative languages
Result language
angličtina
Original language name
A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type
Original language description
Given a bounded measurable function sigma on R-n, we let T-sigma be the operator obtained by multiplication on the Fourier transform by sigma. Let 0 < s(1) <= s(2) <= ... <= s(n) < 1 and psi be a Schwartz function on the real line whose Fourier transform (psi) over cap is supported in [-2, -1/2] boolean OR [1/2, 2] and which satisfies Sigma(j is an element of Z) (psi) over cap (2(-j) xi) = 1 for all xi not equal 0. In this work we provide a sharp form of the Marcinkiewicz multiplier theorem on L-p by finding an almost optimal function space with the property that, if the function (xi(1), ... , xi(n)) -> Pi(n)(i=1)(I - partial derivative(2)(i))(si/2) [Pi(n)(i=1)(psi) over cap(xi(i))sigma(2(j1) xi(1), ... , 2(jn) xi(n))] belongs to it uniformly in j(1), ... , j(n) is an element of Z, then T-sigma is bounded on L-p(R-n) when vertical bar 1/p - 1/2 vertical bar < s(1) and 1 < p < infinity. In the case where s(i) not equal s(i+1) for all i, it was proved in [12] that the Lorentz space L-1/s1,L-1(R-n) is the function space sought. Here we address the significantly more difficult general case when for certain indices i we might have s(i) = s(i+1). We obtain a version of the Marcinkiewicz multiplier theorem in which the space L-1/s1,L-1 is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among s(2), ... , s(n) that equal s(1). Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space. (c) 2021 Elsevier Inc. All rights reserved.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Functional Analysis
ISSN
0022-1236
e-ISSN
1096-0783
Volume of the periodical
282
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
36
Pages from-to
109295
UT code for WoS article
000715370800001
EID of the result in the Scopus database
2-s2.0-85118526392