Abstract
We consider weighted estimates for two bilinear fractional integral operators \(I_{2,\alpha }\) and \(BI_{\alpha }\). Moen (Collect Math 60:213–238, 2009) obtained a necessary and sufficient condition for \(I_{2,\alpha }\). However we know only some sufficient conditions for \(BI_{\alpha }\) which is a variant of the bilinear Hilbert transform. Restricted to power weights we obtain a necessary and sufficient condition for \(BI_{\alpha }\). We also prove a bilinear Stein–Weiss inequality.
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Appendix
Appendix
We show how to use the interpolation theorem (Lemma 3) to obtain (9):
from weak type estimates (10).
Let A, B and C be fixed such that \(A+B+C \ge 0\). For the sake of simplicity, we consider in the following region:
where \(\alpha < n/2\); see Remarks in Sect. 2.
Let \(P:=(1/p,1/q,1/r) \in {\mathbb {D}}\) be fixed, and assume that
It suffices to find three points \(P_i :=(1/p_i,1/q_i,1/r_i) \in {\mathbb {D}}, i=1,2,3\) such that P lies in the interior of the triangle with vertices \(P_1, P_2\) and \(P_3\), and
Let \(p_1=p\), and we take \(q_1 >q\) sufficiently near q such that \(B< n/q_1'\), and \(C< n/r_1\). Similarly let \(q_2=q\) and we take \(p_2>p\) sufficiently near p such that \(A < n/p_2'\) and \(C< n/r_2\). Finally we take \(p_3 < p\) and \(q_3 < q\) sufficiently near p and q respectively such that \(A< n/p_3' , B < n/q_3'\), and \(C< n/r_3\).
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Komori-Furuya, Y. Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights. Collect. Math. 71, 25–37 (2020). https://doi.org/10.1007/s13348-019-00246-5
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DOI: https://doi.org/10.1007/s13348-019-00246-5