Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions

[Sur les inégalités $L\;{\rm log}^\alpha L$ exactes pour les fonctions conjuguées]

Essén, Matts ; Shea, Daniel F. ; Stanton, Charles S.

Annales de l'Institut Fourier, Tome 52 (2002), p. 623-659 / Harvested from Numdam

Résumé

Nous donnons une méthode pour la construction des fonctions $\phi $ et $\psi $ telles que $H(x,y)=\phi (x)-\psi (y)$ aî t une minorante sousharmonique spécifiée. D’après un théorème de B. Cole, ce procédé établit des inégalités d’intégrales pour les fonctions conjuguées. Nous appliquons cette méthode pour déduire des inégalités optimales pour les conjuguées des fonctions de la classe $L\log ^\alpha L$, pour $-1\le \alpha <\infty $. En particulier, le cas $\alpha =1$ procure une amélioration de la version de Pichorides de l’inégalité classique de Zygmund pour les conjuguées des fonctions de $L\log L$. Nous appliquons aussi cette méthode pour obtenir une nouvelle preuve de l’inégalité de M. Riesz pour les fonctions de $L^p$ ($1<p<2$), avec meilleure constante. Toutes ces inégalités sont des cas spéciaux d’une inégalité générale et optimale pour les fonctions conjuguées (cf. Théorème 6).

Abstract

We give a method for constructing functions $\phi $ and $\psi $ for which $H(x,y) = \phi (x)-\psi (y)$ has a specified subharmonic minorant $h(x,y)$. By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class $L\log ^{\alpha }L$, for $-1\le \alpha <\infty $. In particular, the case $\alpha = 1$ yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in $L\log L$. We also apply the method to produce a new proof of the M. Riesz’s inequality for functions in $L^p$, $(1<p<2)$, also with sharp constant. All these inequalities are special cases of a general sharp inequality for conjugate functions (cf. Theorem 6).

Détail

Publié le : 2002-01-01

Zbl 1053.42012

DOI : https://doi.org/10.5802/aif.1896
Classification:  42A50,  30D55,  31A15
Mots clés: fonctions conjuguées, estimation des normes, effilement minimal

Bibliographie

[1] H. Aikawa; M. Essen Potential Theory - Selected Topics, Springer Lecture Notes in Mathematics, Tome 1633 (1996) | MR 1439503 | Zbl 0865.31001

[2] R. Banuelos; G. Wang Sharp inequalities for martingales with applications to the Beurling--Ahlfors and Riesz transforms, Duke Math J., Tome 80 (1995), pp. 575-600 | MR 1370109 | Zbl 0853.60040

[3] D. L. Burkholder An elementary proof of an inequality of R. E. A. C. Paley, Bull. London Math. Soc., Tome 17 (1985), pp. 474-478 | Article | MR 806015 | Zbl 0566.46014

[4] D. L. Burkholder Explorations in Martingale Theory and its applications, Springer Lecture Notes in Mathematics, Tome 1464 (1989), pp. 1-66 | Article | MR 1108183 | Zbl 0771.60033

[5] M. Essén The $\cos\pi\lambda$ Theorem, Springer Lecture Notes in Mathematics, Tome 467 (1975) | MR 466587 | Zbl 0335.31001

[6] M. Essén A superharmonic proof of the M. Riesz conjugate function theorem, Ark. Mat., Tome 22 (1984), pp. 241-249 | Article | MR 765412 | Zbl 0562.30002

[7] M. Essén Harmonic majorization, harmonic measure and minimal thinness, Springer Lecture Notes in Mathematics, Tome 1275 (1987), pp. 89-112 | Article | MR 922294 | Zbl 0671.31002

[8] M. Essén Some best constant inequalities for conjugate functions, Birkhäuser-Verlag, Basel (International Series of Numerical Math.) Tome 103 (1992), pp. 129-140 | Zbl 0771.42004

[9] M. Essén; D. F. Shea; And C. S. Stanton A value-distribution criterion for the class $L \log L$ and some related questions, Ann. Inst. Fourier, Grenoble, Tome 35 (1985) no. 4, pp. 127-150 | Article | Numdam | MR 812321 | Zbl 0563.30025

[10] M. Essén; D. F. Shea; And C. S. Stanton; (Ed. R. P. Agarwal) Some best constant inequalities of $L(\log L)^\alpha$ type, World Scientific Publishing (Inequalities and Applications) Tome 3 (1994), pp. 233-239 | Zbl 0880.30005

[11] M. Essén; D. F. Shea; And C. S. Stanton Near Integrability of the conjugate function, Complex Variables, Tome 37 (1998), pp. 179-183 | MR 1687876 | Zbl 1054.42500

[12] M. Essén; D. F. Shea; And C. S. Stanton Best constant inequalities for conjugate functions, J. Comput. Appl. Math., Tome 105 (1999), pp. 257-264 | Article | MR 1690592 | Zbl 0944.42009

[13] M. Essén; D. F. Shea; And C. S. Stanton; (Ed. Heinonen Kilpeläinen And Koskela) Best constants in Zygmund's inequality for conjugate functions, A volume dedicated to Olli Martio on his 60th birthday, Department of Mathematics, University of Jyväskylä, Tome 83 (2001), pp. 73-80 | Zbl 1051.42007

[14] T. W. Gamelin Uniform algebras and Jensen measure, Cambridge University Press, London Math. Soc. Lecture Note Series, Tome 32 (1978) | MR 521440 | Zbl 0418.46042

[15] W. K. Hayman; P. B. Kennedy Subharmonic functions I, Academic Press (1976) | Zbl 0419.31001

[16] T. Iwaniec; G. Martin Riesz transforms and related singular integrals, J. Reine Angew. Math., Tome 473 (1996), pp. 25-57 | MR 1390681 | Zbl 0847.42015

[17] T. Iwaniec; A. Lutoborskii Integral estimates for null Lagrangians, Arch. Rational Mech. Anal., Tome 125 (1993), pp. 25-79 | Article | MR 1241286 | Zbl 0793.58002

[18] S. K. Pichorides On the best value of the constants in the theorems of M. Riesz, Zygmund, and Kolmogorov, Studia Math., Tome 44 (1972), pp. 165-179 | MR 312140 | Zbl 0238.42007

[19] M. Riesz Sur les fonctions conjugées, Math Z., Tome 27 (1927), pp. 218-244 | Article | JFM 53.0259.02 | MR 1544909

[20] P. Stein On a theorem of M. Riesz, J. London Math. Soc., Tome 8 (1933), pp. 242-247 | Article | Zbl 0007.35003

[21] I. E. Verbitsky An estimate of the norm in a Hardy space in terms of the norms of its real and imaginary parts (Mat. Issled. Vyp.) Tome 54 (1980), pp. 16-20 | Zbl 0557.30027

[21] I. E. Verbitsky An estimate of the norm in a Hardy space in terms of the norms of its real and imaginary parts, Amer. Math. Soc. Transl. (2), Tome 124 (1984), pp. 11-15 | Zbl 0557.30027

[22] A. Zygmund Sur les fonctions conjugées, Fund. Math., Tome 13 (1929), pp. 284-303 | JFM 55.0751.02

[23] A. Zygmund Trigonometric Series, Cambridge University Press (1968) | MR 236587 | Zbl 0085.05601