On Hardy's inequality and Paley's gap theorem (Q1813972)

Let \(\lambda\) denote normalized Lebesgue measure on the circle \(\mathbb{T}\), and for \(f\in L^ 1(\mathbb{T})\) set \(\hat f(n):=\int_{\mathbb{T}} z^{- n}f(z)d\lambda(z), n\in \mathbb{Z}\). Recall that for \(1\leq p<\infty\) the (Hardy) space \(H^ p(\mathbb{T})[H_ 0^ p(\mathbb{T})]\) consists of the \(f\in L^ p(\mathbb{T})\) having \(\hat f(n)=0\) for all \(n<0\) \([n\leq 0]\). Two well- known classical results affirm that (Hardy): \[ \sum_{n=1}^ \infty n^{-1}| \hat f(n)| \leq C_ 1\| f\| _ 1\hbox{ for all }f\in H^ 1(\mathbb{T}),\leqno(1) \] where \(C_ 1\) is a finite constant, and that (Paley) for any Hadamard gap sequence \(\{n_ k\}\) in \(\mathbb{N}\) there is a finite constant \(C_ 2\) satisfying \[ \sum_{k=1}^ \infty | \hat f(n_ k)| ^ 2\leq C_ 2^ 2\| f\| _ 1^ 2\hbox{ for all }f\in H^ 1(\mathbb{T}).\leqno(2) \] The authors generalize these results, in the context of \(\mathbb{T}\) and beyond to locally compact spaces. Let \(\alpha\) be a unimodular Borel function on \(\mathbb{T}\), \(f\in L^ 1(\mathbb{T})\) and write \(\alpha^* f\) for the image measure \((f\cdot \lambda)\circ\alpha^{-1}\). The authors' first generalization of (1) and (2) is: \[ \sum_{n=1}^ \infty n^{-1}\| (\alpha^*\beta^ n)*f\| _ 1\leq C_ 1\| f\| _ 1\hbox{ for all }f\in H_ 0^ 1(\mathbb{T})\leqno(1)' \] and all \(\alpha,\beta\in H^ 1(\mathbb{T})\) satisfying \(| \alpha|=1\geq | \beta|\) a.e., and \(\hat \alpha(0)\hat \beta(0)=0\). Here \(C_ 1\) is the same constant as in (1). If \(\{n_ k\}\) is a Hadamard gap sequence in \(\mathbb{N}\) and \(C_ 2\) is any constant satisfying (2), then \[ \sum_{k=1}^ \infty \| (\alpha^*\beta^{n_ k})*f\| _ 1^ 2\leq C_ 2^ 2\| f\|_ 1^ 2\hbox{ for all }f\in H_ 0^ 1(\mathbb{T}).\leqno(2)' \] The proofs of \((1)'\) and \((2)'\) are similar and utilize the following elementary duality lemma. Let \(G\) be a locally compact space, \(A\) a linear subspace of \(C_ 0(G)\), \(A^ \perp\) the measures in \(M(G)\) which annihilate \(A\), \(\{\gamma_ k\}_{k\geq 1}\) a sequence in \(C_ 0(G)\), \(\{a_ k\}_{k\geq 0}\) a sequence of positive reals and \(1\leq p<\infty\). Consider (a) \(\sum_{k=1}^ \infty a_ k| \langle \gamma_ k,\mu\rangle|^ p\leq a_ 0^ p\| \mu\| ^ p\hbox{ for all }\mu\in A^ \perp\); and (b) \(\hbox{Inf}\{\| \sum_{k=1}^ \infty a_ kc_ k\gamma_ k+\phi\| _ \infty: \phi\in A\}\leq a_ 0\) for all finitely-nonzero complex sequences \(\{c_ k\}\) such that either \(p=1\) and \(\sup_ k| c_ k|\leq 1\) or \(p>1\) and \(\sum a_ k| c_ k|^{p/(p- 1)}\leq 1\). Then (a) and (b) are equivalent. Let \(X\), \(Y\) be two more locally compact spaces. For a Borel mapping \(u: G\times X\to Y\), \(\nu\in M(G)\) and \(\mu\in M(X)\) define an abstract convolution by \(\int\phi d(\nu*_ u \mu):=\int \phi(u(g,x))d\nu(g)d\mu(x)\). The final generalization of (1) and (2) now reads: If either of the equivalent conditions (a) or (b) holds and if \((\phi\nu)*_ u\mu=0\) for all \(\phi\in A\), then \[ \sum_{k=1}^ \infty a_ k\| (\gamma_ k\nu)*_ u\mu\| ^ p\leq a_ 0^ p\| \nu\| ^ p\| \mu\| ^ p. \]