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.
Photonic RF and microwave fractional differentiation, integration, and Hilbert transforms with Kerr optical micro-combs
