Abstract :
Let n 2, Sn−1 be the unit sphere in Rn. For 0 α < 1, m ∈ N0, 1 < p 2, and Ω ∈ L∞(Rn) ×
Hr (Sn−1) with r >
p (n−1)
n+2(α+m) (where Hr is the Hardy space if r 1 and Hr = Lr if 1 < r <∞), we
study the singular integral operator, for r 1, defined by
Tα,mf (x) := p.v. Rn
Ω(x, y)f (x − y)
|y|n+α+m+iω
dy,
where ω ∈ R, f ∈ S(Rn). Calderón and Zygmund [A.P. Calderón, A. Zygmund, On singular integrals,
Amer. J. Math. 78 (1956) 289–309] showed that if Ω satisfies the mean zero condition
Sn−1 Ω(x,y )dy = 0, then there is a C >0 such that T0,0f Lp(Rn) C f Lp(Rn) for all f ∈ S(Rn),
where C does not depend on f . In this paper it will be shown that Tα,mf Lp(Rn) C f L
p
α+m(Rn)
for all f ∈ S(Rn) under the assumption that Sn−1 Ω(x,y )P (y )dy = 0 for all spherical polynomials P
of degree m. This result is obtained by exploring certain mixed norm inequalities of the hyper-Hilbert
transform
Hα,mf (x,y ) :=
∞ 0
f (x −ty )− m
[k]=0
1
k!Dkf (x)(−ty )k
t1+(α+m)+iω
dt, where ω ∈ R.
