Almost-Orthogonality Principles for Certain Directional Maximal Functions
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Almost-Orthogonality Principles for Certain Directional Maximal Functions Jongchon Kim1 Received: 1 April 2020 © Mathematica Josephina, Inc. 2020
Abstract We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp L 2 -bounds for these maximal functions when the underlying direction set is equidistributed in Sn−1 . Keywords Directional maximal function · Directional singular integral · Almost-orthogonality principle Mathematics Subject Classification 42B20 · 42B25
1 Introduction This paper is concerned with L 2 -estimates for certain maximal functions associated with a set of direction ⊂ Sn−1 . For Nikodym and Kakeya maximal functions associated with averages over rectangles of bounded eccentricities, L 2 -estimates are classical; see, e.g., [5,7,13,31]. The first maximal function considered in this paper is associated with averages over line segments in a finite set of directions ⊂ Sn−1 : 1 h>0 2h
M f (x) = sup Mv f (x), where Mv f (x) = sup v∈
h
−h
| f (x − vt)|dt.
The second maximal function is a singular integral variant of M . Suppose that m ∈ C ∞ (R \ {0}) satisfies |m (α) (ξ )| ≤ Cα |ξ |−α for all α ≥ 0. We consider a maximal function T associated with the directional singular integral Tv given by Tv f (ξ ) = m(v · ξ ) f (ξ ):
B 1
Jongchon Kim [email protected] Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada
123
J. Kim
T f (x) = sup |Tv f (x)|. v∈
When m(ξ ) = −i sgn(ξ ), Tv is the directional Hilbert transform. We shall denote by H the maximal function T associated with this particular m. The main goal of this paper is to develop almost-orthogonality principles for M and T . They quantify the contribution to the L 2 -operator norm of these maximal operators from different parts of the direction set and facilitate a divide and conquer argument. In R2 , such results for M were obtained by Alfonseca, Soria, and Vargas [2–4]. We develop weaker versions for M and T which work in every dimension. As a corollary, we obtain sharp L 2 -estimates for these maximal operators when is equidistributed. We say that ⊂ Sn−1 is equidistributed if there is 0 < δ < 1 such that is a maximal δ-separated set of points in Sn−1 . In what follows, we denote by ||T || L p (Rn ) the L p -operator norm of an operator T and write A B to indicate that there is an absolute constant C > 0 such that A ≤ C B. Theorem 1.1 Let n ≥ 3. Assume that ⊂ Sn−1 is equidistributed. Then n−2
||M || L 2 (Rn ) (#) 2(n−1)
n−2
and ||T || L 2 (Rn ) (#) 2(n−1) .
Both bounds in Theorem 1.1 are sharp in general. To see this, one may test M and H to the characteristic function of a ball. The sharp upper bound for ||H || L 2 (Rn ) for equidistributed is due to Joonil Kim [24]. Before we discuss earlier results in R2 , we mention a trivial bound. For each v ∈ n−1 S , Mv and Tv are L p (Rn )-bounded for any 1 < p ≤ ∞ and an
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