On the hypothesis $K^*$ of Hardy, Littlewood and Hooley and its relation with discrete fractional operators

In this paper, we revisited the relation between the Hypothesis $K^*$ of Hardy, Littlewood and Hooley and the boundedness of the discrete fractional operator $$ I_{\lambda, k}f(n)=\sum_{m=1}^\infty {f(n-m^k)\over m^{\lambda}}, $$ with $k\in\Bbb N$, $k\ge 2$, in order to obtain that, for every $\varepsilon >0$, and every $2\le r<1/(1-\lambda)$, $$ \int_0^1 |m_{\lambda, k}(\theta)|^{r} |\theta-a|^{( {1-\lambda\over k} +\varepsilon)r-1} d\theta <\infty, $$ uniformly in $a$, where $$ m_{\lambda, k}(\theta)=\sum_{m=1}^\infty {{e^{2\pi i m^k \theta}}\over m^\lambda}. $$ We recall that the Hypothesis $K^*$ is equivalent to the fact that $m\in L^{2k}$, for every $\lambda >1/2$.

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Published 1 January 2008