A singular integral operator related to block spaces (Q1301353)

The authors consider singular integral operators of the form: \[ T_{P_N, h}f(x) = \text{ p.v.}\int_{\mathbb R^n} K(y)f(x-P_N(|y|)y/|y|) dy \] where \(P_N\) is a real polynomial of degree \(N\) on \(\mathbb R\) such that \[ P_0=0\quad \text{and}\quad K(y) =h(|y|)\Omega(y/|y|)/|y|^n. \] Here \(h\in L^\infty(\mathbb R)\) - making the operator `rough', while \(\Omega\) belongs to a certain block space on the unit sphere and has average zero. The block space \(B_q\) in question consists of those functions of the form \(\sum_m c_m b_m\) such that \(c_m\) are scalars and \(b_m\) are \(q\)-blocks, that is, functions supported in surface balls \(Q_m\) such that \(\|b_m\|_q\leq |Q_m|^{-1/{q'}}\). The norm on \(B_q\) is defined as the infimum of \( \sum_m |c_m|[1+(\log|Q_m|)^2]\) where the infimum is taken over all possible representations of the function and \(|Q|\) denotes the surface measure of \(Q\). The main theorem asserts that if \(\Omega\in B_q\) for some \(q>1\) then \( T_{P_N, h}\) is \(L^p\)-bounded, \(1<p<\infty\), with an operator norm that depends on the degree of \(P_N\) but not on its coefficients. The block space condition is strictly weaker than assuming that \(\Omega\) belongs to some Lebesgue class on the sphere. The boundedness condition on \(h\) can also be weakened to a growth condition on its averages.