On hyper-singular integral operators with variable kernels

AbstractLet 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 byTα,mf(x):=p.v.∫RnΩ(x,y)f(x−y)|y|n+α+m+iωdy, 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−1Ω(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‖Lα+mp(Rn) for all f∈S(Rn) under the assumption that ∫Sn−1Ω(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 transformHα,mf(x,y′):=∫0∞f(x−ty′)−∑[k]=0m1k!Dkf(x)(−ty′)kt1+(α+m)+iωdt,whereω∈R. We also obtain the same boundedness result if r⩽1 when Ω is in the distribution space L∞(Rn)×Hr(Sn−1)