Equivalence of norms in one-sided \(H^ p\) spaces.
Let \(f\) be a locally integrable function on \(\mathbb{R}\). The one-sided Hardy-Littlewood maximal functions are defined as \[ M^+f(x)=\sup_{h>0}\frac1h\int_x^{x+h}| f(y)| \,dy \quad\text{and}\quad M^-f(x)=\sup_{y>0}\frac1h\int_{x-h}^{x}| f(y)| \,dy. \] It is well-known that if \(1<q<\infty\) and \(w(x)\geq0\), a necessary and sufficient condition for \[ \int_{-\infty}^\infty M^+f(x)^qw(x)\,dx\leq C(w,q)\int_{-\infty}^\infty| f(x)| ^qw(x)\,dx \] to hold is that there exists a constant \(C<\infty\) such that if \(-\infty<a<b<c<\infty\), then \[ \bigg(\int_a^b w(x)\,dx\bigg)^{1/q}\bigg(\int_b^c w(x)^{-q'/q}\,dx\bigg)^{1/q'}\leq C(c-a).\tag{1} \] A non-negative and measurable function is a weight in the class \(A_q^+\) if (1) holds. Similar results are valid for \(M^-f\). Let \(\mathcal D\) be the space of the \(C^\infty\) functions with compact support defined on \(\mathbb{R}\). The dual space of \(\mathcal D\) is denoted by \(\mathcal D'\). Let \(S\) be the space of \(C^\infty\) functions on \(\mathbb{R}\) with rapidly decreasing derivatives of all orders. For any \(c\in\mathbb{R}\) let \(S_c\) be the closed subspace of \(S\) of all rapidly decreasing functions with support contained in \([c,+\infty)\). The dual space of \(S_c\) is denoted by \(S_c'\). If \(-\infty\leq d<\infty\) then \(S_d^+\) is defined as the union \(S_d^+=\bigcup_{d<c}S_c\). The set of linear functionals of \(S_d^+\) that restricted to \(S_c\) belong to \(S_c'\), for any number \(c\) greater than \(d\), is denoted by \((S_d^+)'\). Let \(\gamma\) be a non-negative integer. Then \(\varphi\) belongs to the class \(\Phi_\gamma^+\) if \(\varphi\in\mathcal D\), \(\text{supp}(\varphi)\subseteq I\), where \(I\) is a bounded interval of the form \([-a,0)\), \(a>0\), and \(| I| ^{\gamma+1}\| {\mathcal D}^\gamma\varphi\| _\infty\leq1\). If \(f\) belongs to \((S_d^+)'\) then the maximal function \(f_{+,\gamma}^*\) of \(f\) is defined as \(f_{+,\gamma}^*(x)=\sup_{\varphi\in\Phi_\gamma^+}| \langle f,\varphi(x-\cdot)\rangle| \) for \(x>d\). Given a weight \(w(x)\) in the class \(A_q^+\) and \(0<p<\infty\), it is said that \(f\in{\mathcal D}'(x_{-\infty},+\infty)\) belongs to \(H_{+,\gamma}^p(w)\) if \[ \bigg(\int_{x_{-\infty}}^\infty f_{+,\gamma}^* (x)^pw(x)\,dx\bigg)^{1/p}= \| f\| _{H_{+,\gamma}^p(w)}<\infty. \] One of the main results is the following: Let \(w\) belong to \(A_q^+\) and \(0<p<\infty\). If \(\beta\) and \(\gamma\) are positive integers satisfying \(p(\beta+1)>q\) and \(p(\gamma+1)>q\), then \[ H_{+,\beta}^p(w)=H_{+,\gamma}^p(w). \] Moreover, for any distribution \(f\in {\mathcal D}'(x_{-\infty},+\infty)\), \[ c_1\| f\| _{H^p_{+,\beta}(w)}\leq \| f| _{H^p_{+,\gamma}(w)}\leq c_2\| f\| _{H^p_{+,\beta}(w')} \] holds with constants \(c_1\) and \(c_2\) not depending on \(f\). Also some ,other theorems about equivalent norms are given.
scientific article; zbMATH DE number 1780849