Principal values for the signed Riesz kernels of non-integer dimension

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Abstract

For positive measures μ in R n and 0 < α < 1, we study the μ-almost everywhere existence of the principal values of the α-Riesz transform of μ, lime→0∫Ιy-xΙ>ε y-x/Ιy-xΙ1+αdμ(y). We show that the L 2 (μ)-boundedness of the a-Riesz transform implies the existence of the above principal value for μ-almost all x &isin; R n . We also prove that if μ has positive and finite upper density μ-almost everywhere, then the set where the principal value does not exist has positive μ-measure. © 2011 Rocky Mountain Mathematics Consortium.
Original languageEnglish
Pages (from-to)869-883
JournalRocky Mountain Journal of Mathematics
Volume41
Issue number3
DOIs
Publication statusPublished - 8 Sep 2011

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