Weighted estimates for singular integrals via Fourier transform estimates
We consider weighted norm inequalities for singular integral operators \(Tf(x)=p.\nu.f*k(x)\), for kernels k which satisfy no smoothness estimates. Among the results we obtain is the following: If \(1<r\leq \infty\) and \(k(x)=h(| x|)\Omega (x/| x|)/| x|^ n,\) where \(\Omega \in L^ r(S^{n-1})\) has mean value 0, and \(\int^{2R}_{R}| h(t)|^ rdt\leq CR\) uniformly in \(R>0\), then T is bounded on \(L^ p(w)\) provided that either (A) \(p>r'\) and \(w\in A_{p/r'}\), or (B) \(1<p<r\) and \(w^{-1/(p-1)}\in A_{p/r'}\), or (C) \(1<p<\infty\) and \(w^{r'}\in A_ p\). In particular, letting \(h=1\) gives the standard weighted norm inequalities for homogeneous singular operators without the usual smoothness requirements on \(\Omega\). We prove our results by deriving Fourier transform estimates for the kernel and showing that these estimates enable us to approximate T by operators whose kernels satisfy appropriate smoothness conditions, by a procedure similar to that used in the unweighted setting to derive norm inequalities for Hilbert transforms along curves. In a subsequent paper (to appear), the author derives the corresponding estimates for the corresponding maximal singular integral operator and related maximal operators.
scientific article; zbMATH DE number 4169360
scientific article