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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
Zbl 0914.46034
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