Sharp L log α L inequalities for conjugate functions
[Sur les inégalités L log α L exactes pour les fonctions conjuguées]
Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 623-659.

Nous donnons une méthode pour la construction des fonctions φ et ψ telles que H(x,y)=φ(x)-ψ(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 Llog α L, pour -1α<. En particulier, le cas α=1 procure une amélioration de la version de Pichorides de l’inégalité classique de Zygmund pour les conjuguées des fonctions de LlogL. 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).

We give a method for constructing functions φ and ψ for which H(x,y)=φ(x)-ψ(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 Llog α L, for -1α<. In particular, the case α=1 yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in LlogL. 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).

DOI : 10.5802/aif.1896
Classification : 42A50, 30D55, 31A15
Keywords: conjugate functions, norm estimates, minimal thinness
Mot clés : fonctions conjuguées, estimation des normes, effilement minimal
Essén, Matts 1 ; Shea, Daniel F. 2 ; Stanton, Charles S. 3

1 University of Uppsala, Department of Mathematics, Box 480, 751 06 Uppsala (Suède)
2 University of Wisconsin, Department of Mathematics, Madison WI 53706-1313 (USA)
3 California State University at San Bernardino, Department of Mathematics, San Bernardino CA 92407 (USA)
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Essén, Matts; Shea, Daniel F.; Stanton, Charles S. Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 623-659. doi : 10.5802/aif.1896. https://aif.centre-mersenne.org/articles/10.5802/aif.1896/

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

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

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

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

[5] M. Essén The cosπλ Theorem, Springer Lecture Notes in Mathematics, Volume 467 (1975) | MR | Zbl

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

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

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

[9] M. Essén; D. F. Shea; and C. S. Stanton A value-distribution criterion for the class LlogL and some related questions, Ann. Inst. Fourier, Grenoble, Volume 35 (1985) no. 4, pp. 127-150 | DOI | Numdam | MR | Zbl

[10] M. Essén; D. F. Shea; and C. S. Stanton; (ed. R. P. Agarwal) Some best constant inequalities of L(logL) α type (Inequalities and Applications), Volume 3 (1994), pp. 233-239 | Zbl

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

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

[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, Volume 83 (2001), pp. 73-80 | Zbl

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

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

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

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

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

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

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

[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.), Volume 54 (1980), pp. 16-20 | Zbl

[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), Volume 124 (1984), pp. 11-15 | Zbl

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

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

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