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I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q < \infty $, and $\mu = 1 + \frac{1}{q} - \frac{1}{p}$.
For the integral \begin{equation} I(x) = \int_{\mathbb{R}} \frac{f(y) }{|y - x|^{\mu}} \, dy \end{equation} there is the Hardy-Littlewood-Sobolev inequality which states that \begin{equation} ||I||_{L_q(\mathbb{R})} \leq C \, ||f||_{L_p(\mathbb{R})} \end{equation} I would like to know whether there are any similar results for $I^*(x)$.

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  • $\begingroup$ Scaling considerations forces the relationship between $p,q,\mu$ in HLS; since log corrections are small compared to scaling, this suggests that the only possible inequality of the form $\|I^*\|_q \lesssim \|f\|_p$ would have the same $p,q,\mu$ relationship. However, if I am not mistaken the kernel $|x|^{-\mu} \ln |x|$ is not weak $L^{1/\mu}$, so I am pessimistic there. // What kinds of right hand sides do you imagine using for your desired inequality? $\endgroup$ Jun 13, 2017 at 21:05
  • $\begingroup$ I suspect $C$ depends on $p$. $\endgroup$
    – Lewi_Sol
    Jun 13, 2017 at 21:54

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