A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Popis výsledku v původním jazyce

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 &lt; s(1) &lt;= s(2) &lt;= ... &lt;= s(n) &lt; 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)) -&gt; 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 &lt; s(1) and 1 &lt; p &lt; 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.

Název v anglickém jazyce

A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Popis výsledku anglicky

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 &lt; s(1) &lt;= s(2) &lt;= ... &lt;= s(n) &lt; 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)) -&gt; 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 &lt; s(1) and 1 &lt; p &lt; 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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Functional Analysis

ISSN

0022-1236

e-ISSN

1096-0783

Svazek periodika

282

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

36

Strana od-do

109295

Kód UT WoS článku

000715370800001

EID výsledku v databázi Scopus

2-s2.0-85118526392