Two discrete fractional integral operators revisited (Q1809924)

For the operator \(I_\lambda f(m)= \sum^\infty_{n=1} f(m- n^2)n^\lambda\) the norm estimate \(\|I_\lambda f\|_{\ell^p}\leq A\|f\|_{\ell^q}\) is proved for certain \(\lambda\), \(p\), \(q\), in particular, for \(\lambda= 1+ i\gamma\) and \(p= q\). Also a two-dimensional generalization is treated.