Majorization of regular measures and weights with finite and positive critical exponent (Q2465817)

Let \({\mathcal M}^+(\mathbb{R})\) denote the set of all nonnegative, finite Borel measures on \(\mathbb{R}\), and let \[ {\mathcal M}^*(\mathbb{R}):\Biggl\{\mu\in{\mathcal M}^+(\mathbb{R}): \Biggl|\int_{\mathbb{R}} x^nd\mu\Biggr|<+\infty,\;\forall n\geq 0\Biggr\}. \] Let \({\mathcal P}= \mathbb{R}[x]\), let \(\overline{\mathcal P}\) be the set of all topological linear spaces in which \({\mathcal P}\) is dense, and let \({\mathcal M}^*_p(\mathbb{R})= \{\mu\in{\mathcal M}^*(\mathbb{R}); L_p(\mathbb{R}, d\mu)\in \overline{\mathcal P}\}\), \(p> 0\); \({\mathcal M}^*_0(\mathbb{R})={\mathcal M}^*(\mathbb{R})\); and \({\mathcal M}^*_\infty(\mathbb{R})= \bigcap_{p> 0}{\mathcal M}^*_p(\mathbb{R})\). Similarly, let \({\mathcal W}^*(\mathbb{R})\) be the set of upper-semicontinuous functions \(w: \mathbb{R}\to\mathbb{R}^+\), with \(\| x^n\|_w<+\infty\), where \(\| f\|_w= \sup\{w(x)|f(x)|: x\in\mathbb{R}\}\); let \(C^0_w\) be the set of all continuous real functions for which \(\lim_{|x|\to+\infty}\,w(x) f(x)= 0\); and let \({\mathcal W}^*_\tau(\mathbb{R})= \{w\in{\mathcal W}^*(\mathbb{R}): C^0_w\in\overline{\mathcal P}\}\), \(\tau> 0\); \({\mathcal W}^*_\infty(\mathbb{R})={\mathcal W}^*(\mathbb{R})\); and \({\mathcal W}^*_0(\mathbb{R})= \bigcap_{\tau> 0}{\mathcal W}^*_\tau(\mathbb{R})\). The study of polynomial density in the spaces \(L_p(\mathbb{R},d\mu)\) and \(C^0_w\) has a long history for which the authors have provided a graceful, carefully-organized outline. The problem of finding conditions on \(\mu\in{\mathcal M}^*(\mathbb{R})\) under which \(p\) will belong to \({\mathcal M}_p(\mathbb{R})\), for a given \(p\in[1, +\infty)\), was first considered by \textit{C. Berg} and \textit{J. P. R. Christensen} [Ann. Inst. Fourier 31, No. 3, 99--114 (1981; Zbl 0437.42007) and C. R. Math. Acad. Sci., Paris, Sér. I, 296 661--663 (1983; Zbl 0531.28008)], where these authors also began an investigation of the relations between this problem and the moment problem. In the second of those articles they introduced the notion of the critical exponent, which the present authors employ in the current work. The critical exponent, \(\rho(\mu)\), of \(\mu\in{\mathcal M}^*(\mathbb{R})\), is given by \(\rho(\mu)= \{p\in [0,+\infty): \mu\in{\mathcal M}^*_p(\mathbb{R})\}\). The exponent index \(\iota(\mu)\) is \(1\), if \(\mu\in{\mathcal M}^*_{\rho(\mu)}(\mathbb{R})\), and is \(0\) in the contrary case. In similar fashion, the critical exponent of \(w\in{\mathcal W}^* (\mathbb{R})\) is given by \(\rho(w)= \text{inf}\{\tau\in[0,+\infty): w\in{\mathcal W}^*_\tau(\mathbb{R})\}\), and the corresponding exponent index by \(\iota(w)= 1\), if \(w\in{\mathcal W}^*_{\rho(w)}(\mathbb{R})\), and is, otherwise, \(0\). To the question of the existence of measures and weights with given exponents and indices, the authors provide the following answer. For \(w\in{\mathcal W}^*(\mathbb{R})\) and \(\mu\in{\mathcal M}^*(\mathbb{R})\) and \(r> 0\), one defines \(\text{supp\,}w= \{x: w(x)> 0\}\), \(\text{supp\,}p= \{x: \mu((x-\varepsilon, x+ \varepsilon))> 0\), \(\forall\varepsilon> 0\}\), and \(\Lambda^r= \{\{\lambda_k\}_{k\in\mathbb{N}}: \lambda_1\geq 1+ r,\lambda_{k+1}- \lambda_k\geq r\lambda_k\}\). Theorem 1. Given \(\rho\in [0,+\infty]\), \(j\in\{0,1\}\), \(r>0\), and \(\{\lambda_k\}_{k\in\mathbb{N}}\in \Lambda^r\), there exist a weight function \(w\in{\mathcal W}^*(\mathbb{R})\) and a measure \(\mu\in{\mathcal M}^*(\mathbb{R})\) with \(\text{supp\,}w= \text{supp\,}\mu= \{\lambda_k: k\in\mathbb{N}\}\), \(\rho(w)= \rho(\mu)= \rho\), and \(\iota(w)= \iota(\mu)= \chi_{\{0,+\infty\}}(\rho)+ j\chi_{(0,+\infty)}(\rho)\). In order to obtain a constructive description of all weights in \({\mathcal W}^*_1(\mathbb{R})\), it is sufficient to do the same for the weights in a subset, \(X\), of \({\mathcal W}^*_1(\mathbb{R})\), that have a majorizing property; viz., for every \(w\in{\mathcal W}^*_1(\mathbb{R})\) there is an \(w\in X\) such that \(w(x)\leq\omega(x)\), \(\forall x\in\mathbb{R}\). A weight \(w\in{\mathcal W}^*_1(\mathbb{R})\) is regular if \((1+|x|)^nw(x)\in{\mathcal W}^*_1(\mathbb{R})\), \(\forall n\in\mathbb{N}\). In the contrary case, \(w\) is singular. Singular weights are multiplicatively majorizable in \({\mathcal W}^*_1(\mathbb{R})\); indeed, for such a weight, \(w\), there exists an \(n\geq 1\) such that \((1+ |x|)^{n-1} w(x)\in{\mathcal W}^*_1(\mathbb{R})\), while \((1+ |x|)^nw(x)\not\in{\mathcal W}^*_1(\mathbb{R})\). \textit{M. Sodin} [J. Anal. Math. 69, 293--309 (1996; Zbl 0867.41009)] has established the following result concerning the majorization of regular weights: Let \(w\in{\mathcal W}^*_1(\mathbb{R})\) be regular and let \(w_0\in{\mathcal W}^*_1(\mathbb{R})\in{\mathcal W}^*(\mathbb{R})\), if there is a \(\delta> 0\) such that \(\lim_{|x|\to+\infty}\, e^{\delta|x|}w_0(x)= 0\), then \(w+ w_0\in{\mathcal W}^*_1(\mathbb{R})\) and is regular as well. In the present work, the authors show that a regular weight in \({\mathcal W}^*_1(\mathbb{R})\) can be majorized by a regular weight, in the same class, whose critical exponent is 1. Theorem 2. For each regular \(w\in{\mathcal W}^*_1(\mathbb{R})\), there is an \(\omega\in{\mathcal W}^*_1(\mathbb{R})\) such that: (1) \(\omega\) is regular; (2) \(\omega\) majorizes \(w\), and, if \(w\in{\mathcal W}^*_0(\mathbb{R})\), then there are constants \(C_n\in(0,+\infty)\), not depending on \(x\), such that \(w(x)\leq C_n\omega(x)^n\), \(n\geq 2\), \(\forall x\in\mathbb{R}\); (3) \({\mathcal P}\) is not dense in \(C^0_{\omega^\theta}\) for any \(\theta\in(0, 1)\); and (4) \(\rho(\omega)= \iota(\omega)= 1\); i.e., \(w\in{\mathcal W}^*_1(\mathbb{R})\setminus\bigcup_{0< \eta< 1}{\mathcal W}^*_\eta(\mathbb{R})\). A measure \(\mu\in{\mathcal M}^*_p(\mathbb{R})\), \(1\leq p<+\infty\), is \(p\)-regular iff \({\mathcal P}\) is dense in \(L_p(\mathbb{R},(1+ |x|)^{pn}d\mu)\), \(\forall n\in\mathbb{N}\). For \(p\)-regular measures, the authors offer an analogue of Theorem 2; viz., Theorem 3. If \(p\in [1,+\infty)\), and if \(\mu\in{\mathcal M}^*_p(\mathbb{R})\) be \(p\)-regular, then there is a \(\nu\in{\mathcal M}^*(\mathbb{R})\) such that: (1) \(\nu\in{\mathcal M}^*_p(\mathbb{R})\) and is \(p\)-regular; (2) \(\nu\) majorizes \(\mu\); (3) \(\nu\not\in{\mathcal M}^*_q(\mathbb{R})\), \(\forall q> p\); and (4) \({\mathcal P}(nu)= p\), i.e., \(\nu\in{\mathcal M}^*_p(\mathbb{R})\setminus\bigcup_{p< q<+\infty}{\mathcal M}^*_q(\mathbb{R})\).