Abstract
In this note we study the rough singular integral
$$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$
T
Ω
f
(
x
)
=
p
.
v
.
∫
R
n
f
(
x
−
y
)
Ω
(
y
/
|
y
|
)
|
y
|
n
d
y
,
where $n\geq 2$
n
≥
2
and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$
L
log
L
(
S
n
−
1
)
with vanishing integral. We prove that $T_{\varOmega }$
T
Ω
is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$
L
|
x
|
p
L
θ
p
˜
(
R
n
)
, on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$
L
|
x
|
p
L
θ
p
˜
(
R
n
,
ℓ
p
˜
)
and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$
L
p
(
R
n
,
ℓ
p
˜
)
if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$
1
<
p
˜
≤
p
<
p
˜
n
/
(
n
−
1
)
or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $
p
˜
n
/
(
p
˜
+
n
−
1
)
<
p
≤
p
˜
<
∞
. The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.
A note on singular integrals with angular integrability