Two discrete fractional integrals (Q5945781)

\textit{E. M. Stein} and \textit{S. Wainger} [J. Anal. Math. 80, 335--355 (2000; Zbl 0972.42010)] introduced the following analogues of the Hardy-Littlewood fractional integral operators: NEWLINE\[NEWLINEI_{\lambda }(f)(n)=\sum_{m=1}^{\infty }\frac{f(n-m^{2})}{ m^{\lambda }} \quad \text{and} \quad J_{\lambda}(f)(n_{1},n_{2})=\sum_{m=1}^{\infty }\frac{ f(n_{1}-m,n_{2}-m^{2})}{m^{\lambda}}.NEWLINE\]NEWLINE NEWLINEHere it is shown that \( I_{\lambda }\) is bounded from \(l^{p}(Z)\) to \(l^{q}(Z)\) provided that, in addition to the standard requirements that \(0<\lambda<1\) and \(1\leq p<q\leq\infty\), one also has (i) \(1/q<1/p-(1-\lambda)/2\), (ii) \(p<1/(1-\lambda)\) and (iii) \(q>1/\lambda \). NEWLINENEWLINEThis extends the result of Stein and Wainger which also requires \(\lambda>1/2\). The extension is obtained by showing that \( I_{\lambda }\) maps the Lorentz space \(l^{3/2,1}(Z)\) to \(l^{3}(Z)\) whenever \(\lambda=1/3+\varepsilon\) for some \(\varepsilon>0\), then interpolating with a change of \(\lambda\). NEWLINENEWLINESimilar but more intricate techniques are used to extend the Stein-Wainger estimates for \(J_{\lambda}\), again from the range \(1>\lambda>1/2\) to the full range \(0<\lambda<1\), with corresponding restrictions on \(p\) and \(q\). The bounds for \(J_{\lambda}\) involve estimates for the number of ways of representing a pair of integers \((a,b)\) simultaneously as \(a=m_1+m_2+m_3\) and \(b=m_1^2+m_2^2+m_3^2\). Corresponding number-theoretic estimates for higher powers of \(m\) are still open, thus posing an obstacle to extensions of these results to operators like \(I_\lambda^{k}(f)(n)=\sum_{m=1}^{\infty }\frac{f(n-m^{k})}{ m^{\lambda }} \). NEWLINENEWLINEThe author uses direct estimates both for \(I_\lambda\) and \(J_\lambda\) whereas Stein and Wainger used Fourier multiplier estimates. On the other hand, Stein and Wainger were able to use more subtle circle method arguments to obtain bounds for certain endpoints of the range of \(p,q\) that do not follow from the present techniques.