Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

Abstract

We study operators of the form T f ( x ) = ψ ( x ) ∫ f ( γ t ( x ) ) K ( t ) d t , \begin{equation*} Tf(x)= \psi (x) \int f(\gamma _t(x))K(t)\,dt, \end{equation*} where γ t ( x ) \gamma _t(x) is a real analytic function of ( t , x ) (t,x) mapping from a neighborhood of ( 0 , 0 ) (0,0) in R N × R n \mathbb {R}^N \times \mathbb {R}^n into R n \mathbb {R}^n satisfying γ 0 ( x ) ≡ x \gamma _0(x)\equiv x , ψ ( x ) ∈ C c ∞ ( R n ) \psi (x) \in C_c^\infty (\mathbb {R}^n) , and K ( t ) K(t) is a “multi-parameter singular kernel” with compact support in R N \mathbb {R}^N ; for example when K ( t ) K(t) is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth γ t ( x ) \gamma _t(x) , in the single-parameter case when K ( t ) K(t) is a Calderón-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the L p L^p -boundedness of such operators. This paper shows that when γ t ( x ) \gamma _t(x) is real analytic, the sufficient conditions of Street and Stein are also necessary for the L p L^p -boundedness of T T , for all such kernels K K .