\(L_ p\) Markov-Bernstein inequalities for Freud weights
The authors extend to \(L_ p\) the recent \(L_ \infty\) results proved by them. Following is the main result: Theorem. Let \(W:= e^{-Q}\), where \(Q: \mathbb{R}\to\mathbb{R}\) is even and continuous in \(\mathbb{R}\), \(Q(0)= 0\), and \(Q''\) is continuous in \((0,\infty)\), \(Q'(x)\) is positive in \((0,\infty)\), and for some \(A,B>1\), \[ A\leq (x Q'(x))'/Q'(x)\leq B,\quad x\in (0,\infty).\tag{1.1} \] Let \[ \phi_ n(x):= \max\left\{1- {| x|\over a_ n},\;n^{-2/3}\right\},\quad x\in \mathbb{R},\;n\geq 1.\tag{1.2} \] Let \(1\leq p<\infty\). Then there exists \(C>0\) such that for \(n\geq 1\) and \(P\in {\mathcal P}_ n\), \[ \| (PW)' \phi^{-1/2}_ n\|_{L_ p(\mathbb{R})}\leq C {n\over a_ n} \| PW\|_{L_ p(\mathbb{R})}.\tag{1.3} \] (Here \({\mathcal P}_ n\) denotes the class of real polynomials of degree at most \(n\).) Markov-Bernstein inequalities of the form \[ \| P'W\|_{L_ p(\mathbb{R})}\leq C {n\over a_ n'} \| PW\|_{L_ p(\mathbb{R})},\quad P\in {\mathcal P}_ n,\;n\geq 1\tag{1.4} \] have been widely studied and applied in the literature especially in relation to converse theorems of approximation. The essential feature of the theorem is the insertion of the factor \(\phi_ n^{-1/2}\), which is large near \(a_ n\). For \(p=\infty\), the inequality (1.3) was established by the authors [SIAM J. Math. Anal. 21, 1065-1082 (1990; Zbl 0727.41017)], and played an important role in establishing bounds for the orthogonal polynomials for the weight \(W^ 2= e^{-2Q}\) [Constr. Approx. 8, 463-535 (1992; Zbl 0762.41011)]. The case \(A\leq 1\) is also discussed in this paper.
scientific article; zbMATH DE number 636067