On the Hardy and Bellman transforms for orthogonal Fourier series. (Q1809991)

It is well known that if \(f\in L_p[0,\pi]\) with Fourier series \(\sum^\infty_{k=0} a_k\cos kx\), then the series \[ \sum^\infty_{k=0} \Biggl({1\over \kappa+1} \sum^k_{j=0} a_j\Biggr)\cos kx \] is the Fourier series of some function \(Hf\in L_p[0,\pi]\) in case \(1\leq p< \infty\), while the series \[ \sum^\infty_{k=0}\, \Biggl(\sum^\infty_{j=k} {a_j\over j}\Biggr)\cos kx \] is the Fourier series of some function \(Bf\in L_p[0,\pi]\) in case \(1< p\leq\infty\). Furthermore, we have \[ \| Hf\|_p\leq c\| f\|_p\quad\text{and}\quad \| Bf\|_p\leq c\| f\|_p, \] where the constant \(c\) depends on \(p\), but not on \(f\). Now, the present author extends these transformations to the so-called regular orthogonal systems (the notion of regularity is also due to him), and proves the boundedness of these transformations on the Lorentz spaces \(L_{pq}[0,1]\), \(2< p< \infty\), \(1\leq q\leq\infty\).