Lacey, Michael; Thiele, Christoph On Calderón’s conjecture. (English) Zbl 0934.42012 Ann. Math. (2) 149, No. 2, 475-496 (1999). This paper is in continuation of authors’ earlier paper [Ann. Math., II. Ser. 146, No. 3, 693-724 (1997; Zbl 0914.46034)] in which they discussed bilinear operators of the form \[ H_\alpha(f_1,f_2)(x):= \text{p.v. }\int f_1(x- t) f_2(x+\alpha t) dt/t\tag{\(*\)} \] which are originally defined for \(f_1\) and \(f_2\) in the Schwartz class \(S(\mathbb{R})\). The authors investigate whether estimates of the form \[ \|H_\alpha(f_1, f_2)\|_p\leq C_{\alpha,p_1,p_2}\|f_1\|_{p_1}\|f_2\|_{p_2}\tag{\(**\)} \] with constants \(C_{\alpha, p_1,p_2}\) depending only on \(\alpha\), \(p_1\), \(p_2\) and \(p:= p_1p_2/(p_1+ p_2)\) hold. The first result of this type was established in the above cited paper and in this paper the range of the exponents \(p_1\) and \(p_2\) is extended for which the relation \((**)\) is known. One of the theorems proved by the authors is given below:Theorem: Let \(\alpha\in \mathbb{R}\setminus\{0,-1\}\) and \(1<p_1, p_2\leq \infty\), \((2/3)<p:= p_1p_2/(p_1+ p_2)< \infty\), then there exists a constant \(C_{\alpha, p_1,p_2}\) such that estimate \((**)\) holds for all \(f_1,f_2\in S(\mathbb{R})\). Reviewer: Ram Kishore Saxena (Jodhpur) Cited in 11 ReviewsCited in 144 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 44A15 Special integral transforms (Legendre, Hilbert, etc.) 42A50 Conjugate functions, conjugate series, singular integrals 46F12 Integral transforms in distribution spaces Keywords:singular integrals; bilinear operators; Marcinkiewicz interpolation; maximal functions; partial order; Calderón-Zygmund theory; orthogonality Citations:Zbl 0914.46034 PDF BibTeX XML Cite \textit{M. Lacey} and \textit{C. Thiele}, Ann. Math. (2) 149, No. 2, 475--496 (1999; Zbl 0934.42012) Full Text: DOI arXiv EuDML Link OpenURL