Extensions of Hardy-Littlewood inequalities (Q1321873)

Summary: For a function \(f \in H^ p(B_ n)\), with \(f(0)=0\), we prove: If \(0<p \leq s\), then \[ \int^ 1_ 0r^{-1} \left( \log {1 \over r} \right)^{s \beta-1} M^ s_ p(r,R^ \beta f)dr \leq \| f \|_ p^{s-p} \| f \|^ p_{p,s,\beta}. \] If \(s \leq p<\infty\), then \[ \| f \|^ p_{p,s,\beta} \leq \| f \|^{p-s}_ p \int^ 1_ 0r^{-1} \left( \log {1 \over r} \right)^{s\beta-1} M^ s_ p(r,R^ \beta f) dr \] where \(R^ \beta f\) is the fractional derivative of \(f\). These results generalize the known cases \(s=2\), \(\beta=1\) [\textit{G. H. Hardy} and \textit{J. E. Littlewood}, Q. J. Math., Oxf. Ser. 8, 161-171 (1937; Zbl 0017.16203)].