Kakeya-Type Sets Over Cantor Sets of Directions in Rd+1

• Published in: The Journal of fourier analysis and applications Vol. 22; no. 3; pp. 623 - 674
• Main Authors: Kroc, Edward, Pramanik, Malabika
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2016
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Fourier Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Signal,Image and Speech Processing; Cantor sets; 42B25; Secondary 60K35; Primary 28A75; Kakeya sets; Directional maximal operators
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-015-9426-x

Given a Cantor-type subset
Ω
of a smooth curve in
R
d
+
1
, we construct examples of sets that contain unit line segments with directions from
Ω
and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small
(
d
+
1
)
-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of
Ω
, and extends to higher dimensions an analogous planar result of Bateman and Katz (Math Res Lett 15(1):73–81,
2008
). In contrast to the planar situation, a significant aspect of our analysis is the classification of intersecting tube tuples relative to their location, and the deduction of intersection probabilities of such tubes generated by a random mechanism. The existence of these Kakeya-type sets implies that the directional maximal operator associated with the direction set
Ω
is unbounded on
L
p
(
R
d
+
1
)
for all
1
≤
p
<
∞
.