Norm inequalities for certain classes of functions and their Fourier transforms (Q2642198)

For a sequence \(\{\lambda_n\}\), denote by \(\lambda_n^\ast\) the nonincreasing rearrangement of \(| \lambda_n| \). Hardy and Little\-wood showed that for \(q\geq2\), a necessary and sufficient condition that \(\lambda_n\) be, for every rearrangement, the Fourier coefficients of a function \(f\in L^q(0, \pi)\) is that \(\sum(\lambda_n^\ast)^qn^{q-2}<\infty\); and then \[ \| f\| _{L^q(0, \pi)}\leq C\left(\sum(\lambda_n^\ast)^qn^{q-2}\right)^{\frac{1}{q}} \] for every such \(f\). They also showed that if \(f(x)=\sum\lambda_n\cos nx\in L^q\), \(1<q\leq2\), then \(\| \{\lambda_n\}\| _{\ell(q', q)}\leq C\| f\| _{L^q}\), where \(\ell(p,q)\) denotes the Lorentz space of sequences. Conversely, if \(\| \{\lambda_n\}\| _{\ell(q', q)}<\infty\) and \(1<q<\infty\), then \(f\), defined as \(f(x)=\sum\lambda_n^\ast\cos nx\), satisfies \(\| f\| _{L^q}\leq C\| \{\lambda_n\}\| _{\ell(q', q)}\). In this paper, the authors apply tools of interpolation theory and a commutative property of the Hilbert transform to prove necessary and sufficient conditions related to trigonometric series. These results extend and improve related theorems proven by several authors, summarized by Boas. In addition, the authors explore inequalities and operators, both connected to Hardy's inequalities, on certain classes of functions, including quasimonotone functions.