(A’) K(A
µ
x) = |det(A
µ
)|
−1
K(x)
where A
µ
= diag[µ
α
1
, . . . , µ
α
n
] is a diagonal matrix.
For each fixed x ∈ R
n
, the function
F (x, ρ) =
n
X
i=1
x
i
2
ρ
2α
i
is a strictly decreasing function of ρ > 0. Therefore, there exists an unique ρ = ρ(x) for which
F (x, ρ) = 1. It was proved in [6] that ρ is a metric on R
n
. Furthermore, we observe that
(R
n
, ρ) is a homogeneous group that admits a family of dilation δ
µ
= exp(A log µ) for which
ρ(δ
µ
x) = µρ(x), µ > 0, where A is a diagonalizable linear operator with positive eigenvalues.
By the above coordinates of polar type, one now has that
T f(x) = p.v.
Z
R
n
f(x − y)Ω(y
0
)ρ(y)
−α
dy
where Ω(y
0
) = K(y
0
) satisfying
Z
S
n−1
Ω(y
0
)J(y
0
) dσ(y
0
) = 0. (1.1)
The following theorem was proved by Fabes and Riviere in [6]:
Theorem A ([6]) If Ω ∈ C
1
(S
n−1
) and satisfies (1.1), then the operator T is bounded on
L
p
(R
n
) for 1 < p < ∞.
Later, the above theorem was improved by Nagel, Riviere and Wainger [9] and the regularity
condition assumed on Ω was removed.
Theorem B ([9]) If Ω ∈ L log
+
L(S
n−1
) and satisfies (1.1), then the operator T is bounded
on L
p
(R
n
) for 1 < p < ∞.
On the other hand, it was known in [10] that in a special case α
1
= α
2
= · · · = α
n
= 1,
the condition Ω ∈ L log
+
L(S
n−1
) can be replaced further by a weaker condition Ω ∈ H
1
(S
n−1
),
where H
1
(S
n−1
) is the Hardy space that contains L log
+
L(S
n−1
) as a proper subspace on the unit
sphere. Thus a natural question is if the conclusion of Theorem B holds still when replacing
Ω ∈ L log
+
L(S
n−1
) by Ω ∈ H
1
(S
n−1
). The main purpose of this paper is to give a positive
answer to the above question.
Theorem 1 If Ω ∈ H
1
(S
n−1
) and satisfies (1.1), then the operator T is bounded on L
p
(R
n
)
for 1 < p < ∞.
Before proving the theorem, we want to say a few words. First, we understand that the
underlying space is a special homogeneous group (see [8]). Thus many standard results might
be adapted. For instance, we can use the Littlewood-Paley theory without any modifications.
Second, though we will follow the ideas in [7], we find that it is not an easy process of copy and
paste. We must obtain some non-trivial estimates in our proof.
The reader can find these new estimates in Section 3. In Section 2 we will present some basic
definitions and known lemmas. The letter C in the paper denotes positive constant independent
of essential variables.
3
http://www.paper.edu.cn
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