Local estimates that lead to weighted estimates for oscillating kernels in two dimensions (Q1118782)

The main results are that, in 2 dimensions, with weights \(w(x)=(1+x^ 2)^{\alpha}\), \(\| K_{a,b+iy}*f\|_{q,w}\leq c(1+| y|^ 2\| f\|_{q,w}\) for \(2\leq q\leq a/(1-b)\) and \(0\leq \alpha \leq bq+a- 2\). Also, estimates, independent of R and v, are obtained, for v in 2 dimensions, for \(| \int_{R}K_{a,b}(t)\exp (-it\cdot v)dt|,\) where R denotes various rectangles.