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Two weight norm inequalities for the bilinear fractional integrals

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Abstract

In this paper, we give a characterization of the two weight strong and weak type norm inequalities for the bilinear fractional integrals in terms of Sawyer type testing conditions. Namely, we give the characterization of the following inequalities,

$$\|\mathcal{I}_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^q(w)} \le \mathscr{N} \prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)}$$

and

$$\|\mathcal{I}_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^{q,\infty}(w)} \le \mathscr{N}_{\rm{weak}}\prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)},$$

when q ≥ p 1, p 2 > 1 and p 1 + p 2 ≥ p 1 p 2.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Kangwei Li & Wenchang Sun

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  1. Kangwei Li
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  2. Wenchang Sun
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Correspondence to Wenchang Sun.

Additional information

This work was partially supported by the National Natural Science Foundation of China (11371200 and 11525104).

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Li, K., Sun, W. Two weight norm inequalities for the bilinear fractional integrals. manuscripta math. 150, 159–175 (2016). https://doi.org/10.1007/s00229-015-0800-4

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