English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let Θ ⊂ S1 be a lacunary set of directions of order D. We show that the maximal directional Hilbert transform $${H_Theta }f(x): = mathop {sup}limits_{v  Theta } |p.vintlimits_mathbb{R} {f(x + tv)frac{{dt}}{t}} |$$obeys the bounds ||HΘ||Lp→Lp ≃p,D (log#Θ) 1/2, for all 1 u003c p u003c ∞. For vector fields vD with range in a lacunary set of order D and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field vD, $${H_{{v_D}}},1f(x): = p.v.int_{|t| leq 1} {f(x + t{v_D}(x))frac{{dt}}{t}} ,$$satisfies the bounds ||({H_{{v_D}}}),1||Lp→Lp ≲p,D 1 for all 1 u003c p u003c ∞. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.

A sharp estimate for the Hilbert transform along finite order lacunary sets of directions