Fractional integral inequality (Hardy-Littlewood)

Question

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 $I^*(x)$.

Answer 1

Not really answering your question but hopefully pointing you in the right direction.

Hardy-Littlewood inequality is a special case of Young's inequality. Young's inequality has been extended to Lorentz spaces in this paper O'Neil, R. O’Neil, Convolution operators and $L(p,q)$ spaces, Duke Math. J. 30 (1963), 129–142. Unfortunately, you need a subscription to access the paper.

This paper, NEW YOUNG INEQUALITIES AND APPLICATIONS instead is accessible Martinez. It has a good bibliography on Young's inequality.

I am thinking of Lorentz spaces $L^{p,q}$ because functions of the type $f(x)=\frac1{|x|^a}$ are not in $L^p$ but in some $L^{p,q}$. Concerning $f(x)=\frac{\log|x|}{|x|^a}$ I have only seen the truncated version $f(x)=\frac{\log|x|}{|x|^a}\chi_{B(0,1/2)}$. It's an exercise in Grafakos, Classical Fourier Analysis Grafakos, Excercise 1.4.8 (only in dimension one).