, Volume 24, Issue 1, pp 375-416

Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane

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Let K be a Calderón–Zygmund convolution kernel on ℝ. We discuss the L p -boundedness of the maximal directional singular integral $$T_{\mathbf{V}} f (x)= \sup_{v \in \mathbf{V}} \bigg| \int_{\mathbb{R}} f(x+t v) K(t) \, \mathrm{d} {t} \bigg| $$ where V is a finite set of N directions. Logarithmic bounds (for 2≤p<∞) are established for a set V of arbitrary structure. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set.

We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L p almost orthogonality principle for Fourier restrictions to cones.

Communicated by Michael Lacey.
The first author is partially supported by a Sloan Research Fellowship and by NSF Grant DMS-0901208. The second author was partially supported by the National Science Foundation under the grant NSF-DMS-0906440, and by the Research Fund of Indiana University.