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On Calderón’s conjecture. (English) Zbl 0934.42012

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 \(f_1,f_2\in S(\mathbb{R})\).

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

Citations:

Zbl 0914.46034
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