On a hyper-Hilbert transform and singular integrals

Let n ≥ 2, Rn be the n-dimensional Euclidean space and Sn-1 be the unit sphere in Rn . For 0 ≤ alpha < 1, m ∈ N0 , 1 < p ≤ 2, p' = pp-1 and O ∈ Linfinity( Rn ) x Hr(Sn -1) with r > p'n-1 n+2a+m (where Hr is the Hardy space if r ≤ 1 and Hr = Lr if 1 < r < infinity), let Talpha,mf (x) := p.v. Rn Wx,x- yf y x-y n+a+m dy .; Remark. The definition of Talpha,m, needs to be modified if O ∈ Linfinity( Rn ) x Hr(Sn -1) with r < 0.; Calderon and Zygmund showed that if O satisfies the condition Sn-1 O(x, y')dy ' = 0, Sn-1 then there is a C > 0 such that T0,0f LpRn for all Schwartz-functions f ∈ SRn , where C does not depend on f.; Under the exact same assumptions, Chen, Ding and Fan extended this result to obtain that Ta,0f LpRn ≤ C fL paRn for all f ∈ SRn .; In this paper it will be shown that for all integers m Ta,mf LpRn ≤ C fL pa+mRn for all f ∈ SRn under the assumption that Sn-1 O(x, y')P( y')dy' = 0 for all spherical polynomials P of degree ≤ m. We note that if m > p'n-1 -n-2a2 , then O ∈ Linfinity ( Rn ) x Hr(Sn -1) is a distribution. Thus the significance of our result is that singular integrals can have a distribution variable kernel.; Our result is obtained by exploring mixed norm inequalities of the Hyper-Hilbert transform Halpha,mf(x, y ') := 0infinityf x-ty'- k=0m 1k!Dkfx -ty' kt1+a+m +iwdt , where o ∈ R , k = (k1, k2,...,k n) ∈ Nn0 .; For the case m = 1, an alternative proof of the boundedness of the operator will be presented, using the rotation method introduced by Calderon and Zygmund.