Singular integrals along lacunary directions in Rn

Abstract A recent result by Parcet and Rogers is that finite order lacunarity characterizes the boundedness of the maximal averaging operator associated to an infinite set of directions in R n . Their proof is based on geometric-combinatorial coverings of fat hyperplanes by two-dimensional wedges. Seminal results by Nagel-Stein-Wainger relied on geometric coverings of n-dimensional nature. In this article we find the sharp cardinality estimate for singular integrals along finite subsets of finite order lacunary sets in all dimensions. Previous results only covered the special case of the directional Hilbert transform in dimensions two and three. The proof is new in all dimensions and relies, among other ideas, on a precise covering of the n-dimensional Nagel-Stein-Wainger cone by two-dimensional Parcet-Rogers wedges.

[1]  I. Parissis,et al.  Directional square functions , 2020, 2004.06509.

[2]  I. Parissis,et al.  A sharp estimate for the Hilbert transform along finite order lacunary sets of directions , 2017, Israel Journal of Mathematics.

[3]  Shaoming Guo.  Hilbert transform along measurable vector fields constant on Lipschitz curves : L2 boundedness , 2015 .

[4]  J. Bourgain.  Analysis at Urbana: A remark on the maximal function associated to an analytic vector field , 1989 .

[5]  R. Fefferman,et al.  On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[6]  A. Carbery.  Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem , 1988 .

[7]  Singular integrals along $N$ directions in $\mathbb{R}^{2}$ , 2010 .

[8]  J. Wilson,et al.  Some weighted norm inequalities concerning the schrödinger operators , 1985 .

[9]  Multi-parameter singular Radon transforms , 2010, 1012.2610.

[10]  K. Rogers,et al.  Directional maximal operators and lacunarity in higher dimensions , 2015 .

[11]  I. Łaba,et al.  On the Maximal Directional Hilbert Transform , 2017, Analysis Mathematica.

[12]  Henri Martikainen.  Representation of bi-parameter singular integrals by dyadic operators , 2011, 1110.1890.

[13]  Michael T Lacey,et al.  Maximal theorems for the directional Hilbert transform on the plane , 2003 .

[14]  On unboundedness of maximal operators for directional Hilbert transforms , 2006 .

[15]  I. Parissis,et al.  On the Maximal Directional Hilbert Transform in Three Dimensions , 2017, International Mathematics Research Notices.

[16]  C. Thiele,et al.  HILBERT TRANSFORMS ALONG LIPSCHITZ DIRECTION FIELDS: A LACUNARY MODEL , 2016, 1603.03317.

[17]  I. Parissis,et al.  Directional square functions and a sharp Meyer lemma. , 2019, 1902.03644.

[18]  Square functions for bi-Lipschitz maps and directional operators , 2017, Journal of Functional Analysis.

[19]  Differentiation in lacunary directions. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Singular Integrals Along N Directions in R^2 , 2010, 1001.1998.

[21]  C. Demeter,et al.  Logarithmic Lp Bounds for Maximal Directional Singular Integrals in the Plane , 2012, 1203.6624.

[22]  P. Sjölin,et al.  Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets , 1981 .

[23]  M. Lacey,et al.  On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields , 2007, 0704.0808.

[24]  D. S. Kurtz.  Littlewood-Paley and multiplier theorems on weighted ^{} spaces , 1980 .