This thesis is concerned with the weighted L2 boundedness of a family of convolution operators on the line with oscillating kernels. It is proved that these convolution operators are bounded from L2(w) to L2(W) where the Borel measures w and W are in a correspondence given by a maximal function and there is a sense in which this maximal function is the best possible. It is also shown that a one-weighted L2 estimate holds for a family of convolution operators with radial oscillating kernels on n-dimensional space
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