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HILBERT TRANSFORMS ALONG LIPSCHITZ DIRECTION FIELDS: A LACUNARY MODEL
Author(s) -
Guo Shaoming,
Thiele Christoph
Publication year - 2017
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579316000280
Subject(s) - lacunarity , lacunary function , mathematics , lipschitz continuity , hilbert transform , lipschitz domain , mathematical analysis , field (mathematics) , pure mathematics , fractal , statistics , fractal dimension , spectral density
We prove bounds for the truncated directional Hilbert transform inL p ( R 2 )for any 1 < p < ∞ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for p = 2 , and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some p .