Research Article
Endpoint Estimates for Oscillatory Singular Integrals with
Hölder Class Kernels
Hussain Al-Qassem ,
1
Leslie Cheng,
2
and Yibiao Pan
3
1
Department of Mathematics and Physics, Qatar University, Doha, Qatar
2
Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA
3
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
Correspondence should be addressed to Hussain Al-Qassem; husseink@qu.edu.qa
Received 29 November 2018; Accepted 1 January 2019; Published 16 January 2019
Academic Editor: Alberto Fiorenza
Copyright © 2019 Hussain Al-Qassem et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We prove the uniform
1
→
1,∞
and
1
→
1
boundedness of oscillatory singular integral operators whose kernels are the
products of an oscillatory factor with bilinear phase and a Calder´ on-Zygmund kernel (,) satisfying a H¨ older condition. Tis
H¨ older condition appreciably weakens the
1
condition imposed in existing literature.
1. Introduction
Let ∈ N. We shall consider the following oscillatory singular
integral operator:
,
: → p.v.∫
R
(,)
(,)() (1)
where (⋅,⋅) is a real-valued bilinear form. In past studies
of this type of operators, (,) is typically assumed to be
a Calder´ on-Zygmund kernel satisfying a
1
condition away
from the diagonal Δ = {(, ) : ∈ R
}, i.e., there exists an
>0 such that
(i) for all (, ) ∈ (R
× R
)\Δ,
(,)
≤
−
;
(2)
(ii) (, ) ∈
1
((R
× R
)\Δ), and for (, ) ∈ (R
×
R
)\Δ
∇
(,)
+
∇
(,)
≤
−
+1
;
(3)
(iii)
2
(R
)→
2
(R
)
≤ (4)
where
() = p.v.∫
R
(,)(). (5)
Under conditions (i), (ii), and (iii), Phong and Stein
proved the
boundedness of
,
for 1<<∞ ([1]). Te
result of Phong and Stein was then extended to operators
with polynomial phases by Ricci and Stein ([2]), under the
same conditions (i), (ii), and (iii) on (,), while the
weak (1,1) boundedness of such operators was subsequently
established by Chanillo and Christ in [3] (for all polynomial
phase functions, bilinear or otherwise).
Te
1
property of in condition (ii) was instrumental
when van der Corput’s lemma, a standard tool in the
treatment of oscillatory integrals, was used in past studies,
including the seminal papers cited above. Tere has been
widespread interest in fnding out what happens when the
1
kernel (,) is replaced by a “rougher” kernel. Many
interesting results have been obtained for kernels that are
homogeneous and of convolutional type but lack smoothness
(i.e., (,)=|−|
−
Ω((−)/|−|)). See, for example,
[4–6].
In this paper we are interested in general kernels (,)
for which condition (ii) is replaced by the following weaker
condition of H¨ older type:
Hindawi
Journal of Function Spaces
Volume 2019, Article ID 8561402, 7 pages
https://doi.org/10.1155/2019/8561402