L2×L2→L1 boundedness criteria

• Published in: Mathematische annalen Vol. 376; no. 1-2; pp. 431 - 455
• Main Authors: Grafakos, Loukas, He, Danqing, Slavíková, Lenka
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2020; Springer Nature B.V
• Subjects: Criteria; Mathematics; Mathematics and Statistics; Operators (mathematics); 42B99; 42B15; 42B20
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-018-1794-5

We obtain a sharp
L
2
×
L
2
→
L
1
boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the
L
q
integrability of this function; precisely we show that boundedness holds if and only if
q
<
4
. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal
L
2
×
L
2
→
L
1
boundedness criterion for bilinear operators associated with multipliers with
L
∞
derivatives. This result provides the main tool in the proof of the first theorem and is also manifested in terms of the
L
q
integrability of the multiplier. The optimal range is
q
<
4
which, in the absence of Plancherel’s identity on
L
1
, should be compared to
q
=
∞
in the classical
L
2
→
L
2
boundedness for linear multiplier operators.