Colorado, E.; Pestana, D.; Rodríguez, J. M.; Romera, E. Muckenhoupt inequality with three measures and applications to Sobolev orthogonal polynomials. (English) Zbl 1316.42031 J. Math. Anal. Appl. 407, No. 2, 369-386 (2013). The aim of the paper is to generalize the classical Muckenhoupt inequality considering three measures instead of two in such a way that for a positive constant \(c\) \(\| f \|_{L^p(\nu_1)} \leq c (\| f \|_{L^p(\nu_2)}+ \| f' \|_{L^p(\nu_3)})\) holds for all regular enough functions \(f\). First, the authors give a sufficient condition on the measures in order to satisfy the generalization of the Muckenhoupt inequality in the following way:“If \(\nu_1, \nu_2\) and \(\nu_3\) are measures on \([a,b]\) and \(\Lambda_{p,b}((\nu_1-k\nu_2)_{+},\nu_3)< \infty\) for some constant \(k \geq 0\), then there exists a constant \(C\) such that \[ \| \int_a^x f(t)dt \|_{L^p([a,b], \nu_1) } \leq c \left( \| \int_a^x f(t)dt \|_{L^p([a,b], \nu_2) }+ \| f \|_{L^p([a,b], \nu_3) } \right) \] for any measurable function on \([a,b]\).”Moreover they give several different hyphoteses for which the previous condition is also necessary and they also show a class of measures which do not satisfy the generalized inequality. As an application to weighted Sobolev spaces, the authors characterize the boundedness of the operator multiplication for a large class of measures, including the most usual examples in the theory of orthogonal polynomials. In this way they obtain a characterization of the boundedness of the zeros and the asymptotic behavior of some Sobolev orthogonal polynomials. Reviewer: Alicia Cachafeiro López (Vigo) Cited in 5 Documents MSC: 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:Muckenhoupt inequality; multiplication operator; zero location; weight, Sobolev orthogonal polynomials; weighted Sobolev spaces PDFBibTeX XMLCite \textit{E. Colorado} et al., J. Math. Anal. Appl. 407, No. 2, 369--386 (2013; Zbl 1316.42031) Full Text: DOI arXiv References: [1] Adams, R. A.; Fournier, J. F., (Sobolev Spaces. Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), vol. 140 (2003), Elsevier/Academic Press: Elsevier/Academic Press Amsterdam) · Zbl 1098.46001 [2] Alvarez, V.; Pestana, D.; Rodríguez, J. 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