@article{f6985718321c41b1ad9d365852cf673b,

title = "The cauchy-leray integral: Counterexamples to the Lp-theory",

abstract = "We prove the optimality of the hypotheses guaranteeing the Lp-boundedness for the Cauchy-Leray integral in Cn, n ≥ 2, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of strong C-linear convexity, and the regularity of order 2, are both necessary.",

keywords = "Cauchy Integral, Cauchy-Leray integral, Cauchy-Szego projection, Hardy space, Lebesgue space, Leray-Levi measure, Minimal smoothness, Pseudoconvex domain",

author = "Loredana Lanzani and Stein, {Elias M.}",

note = "Funding Information: The first and second authors have both been supported in part by the National Science Foundation (award nos. DMS-1503612 and DMS-1265524, respectively). The authors also express their gratitude to the reviewer of this work for several helpful suggestions and comments, in particular for pointing out a duality between two examples: namely, the fact that Example 2 with m = 3 4 is the convex polar of Example 1 (see [APS]). Funding Information: The case a = 0 corresponds to induced Lebesgue measure; the case a = 1, to the Leray-Levi measure dλ; and the case a = 31, to the Fefferman measure [B2], [F-1], [G]. (Here, the expression “A corresponds to B” may take the meaning that A ≈ B.) Acknowledgements. The first and second authors have both been supported in part by the National Science Foundation (award nos. DMS-1503612 and DMS-1265524, respectively). Publisher Copyright: {\textcopyright} Indiana University Mathematics Journal",

year = "2019",

doi = "10.1512/iumj.2019.68.7786",

language = "English (US)",

volume = "68",

pages = "1609--1621",

journal = "Indiana University Mathematics Journal",

issn = "0022-2518",

publisher = "Indiana University",

number = "5",

}