Estimations of Holder Regularities and Direction of Singularity by Hart Smith and Curvelet Transforms

Using Hart Smith's and curvelet transforms, new necessary and new sufficient conditions for an L (2)(R-2) function to possess Holder regularity, uniform and pointwise, with exponent alpha > 0 are given. Similar to the characterization of Holder regularity by the continuous wavelet transform, the conditions here are in terms of bounds of the transforms across fine scales. However, due to the parabolic scaling, the sufficient and necessary conditions differ in both the uniform and pointwise cases. We also investigate square-integrable functions with sufficiently smooth background. Specifically, sufficient and necessary conditions, which include the special case with 1-dimensional singularity line, are derived for pointwise Holder exponent. Inside their "cones" of influence, these conditions are practically the same, giving near-characterization of direction of singularity.