Abstract
We discuss a sharpened Hausdorff–Young inequality and estimate the maximal coefficients of orthogonal expansions in terms of Freud polynomials when \(1<p<2\) and \(2<p<\infty\). We also consider n-dimensional expansions by orthogonal functions associated to Freud-type weights when \(1<p<2\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press (Orlando, Florida, 1988).
P. L. Butzer, The Hausdorff–Young theorems of Fourier analysis and their impact, J. Fourier Anal. Appl., 1 (1994), 113–130.
A. P. Calderón, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math., 26 (1966), 273–299.
C. P. Calderón and A. Torchinsky, Maximal integral inequalities and Hausdorff–Young, submitted for publication.
C. P. Calderón and A. Torchinsky, The Hausdorff–Young inequality for n–dimensional Hermite expansions, Sel. Mat., 9 (2022), 227–233.
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (1998), 409–425.
Z. Ditzian, Expansion by polynomials with respect to Freud-type weights, J. Math. Anal. Appl., 398 (2013), 582–587.
G. Freud, Orthogonal Polynomials, Pergamon Press (1971).
R. A. Hunt, On L(p, q) spaces, Enseign. Math. (2), 12 (1966), 249–276.
M. Jodeit, Jr. and A. Torchinsky, Inequalities for Fourier transforms, Studia Math., 37 (1971), 245–276.
M. A. Krasnosels’kii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Nordhoff (Groningen, 1961).
E. Levin and D. S. Lubinsky, Orthogonal Polynomials for Exponential Weights, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4, Springer- Verlag (New York, 2001).
E. T. Oklander, Interpolación, Espacios de Lorentz, y el Teorema de Marcinkiewicz, Cursos y Seminarios 20, U. de Buenos Aires (1965).
E. T. Oklander, Lpq interpolators and the theorem of Marcinkiewicz, Bull. Amer. Math. Soc., 72 (1966), 49–53.
M. A. Pinsky and C Prather, Pointwise convergence of n-dimensional Hermite expansions, J. Math. Anal. Appl., 199 (1996), 620–628.
B. Simon, Harmonic Analysis: A Comprehensive Course in Analysis, Part 3, Amer. Math. Soc. (Providence, RI, 2015).
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, Princeton University Press (Princeton, NJ, 1971).
A. Torchinsky, Interpolation of operations and Orlicz classes, Studia Math., 59 (1976/77), 177–207.
W. Urbina–Romero, Gaussian Harmonic Analysis, Springer International Publishing (2019).
A. Zygmund, Trigonometric Series, Vols. I, II, 3rd ed., Cambridge Mathematical Library, Cambridge University Press (Cambridge, 2002).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Calderón, C.P., Torchinsky, A. The Hausdorff–Young Inequality and Freud weights. Acta Math. Hungar. 170, 681–703 (2023). https://doi.org/10.1007/s10474-023-01354-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-023-01354-2