Energy conditions and twisted localizations of operators

Abstract

We show that the energy conditions are not necessary for boundedness of fractional Riesz transforms in dimension at least 2. We also give a weak converse, namely that the energy conditions are necessary for boundedness of families of twisted localizations of fractional singular integrals having the positive gradient property.

arXiv comment32 pages. The kernels of the localized operators satisfy only one-sided CZ smoothness estimates, correcting a mistake in the first version. The main result on failure of necessity of energy for Riesz transforms is unaffected

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