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? $\endgroup$– Willie WongJun 13, 2017 at 21:05
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$\begingroup$ I suspect $C$ depends on $p$. $\endgroup$– Lewi_SolJun 13, 2017 at 21:54
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