Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Simon, Péter; Weisz, Ferenc

Studia Mathematica, Tome 122 (1997), p. 231-246 / Harvested from The Polish Digital Mathematics Library

Résumé

Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(\sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} {| f ̂ (k, j) |}^{p} {(k j)}^{p - 2})^{1 / p} \leq C_{p} ∥ f ∥_{H_{* *}^{p}}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{* *}^{p} (G_{m} \times G_{s})$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.

Publié le : 1997-01-01

Zbl 0891.42015

EUDML-ID : urn:eudml:doc:216435

@article{bwmeta1.element.bwnjournal-article-smv125i3p231bwm,
     author = {P\'eter Simon and Ferenc Weisz},
     title = {Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {231-246},
     zbl = {0891.42015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv125i3p231bwm}
}

Simon, Péter; Weisz, Ferenc. Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients. Studia Mathematica, Tome 122 (1997) pp. 231-246. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv125i3p231bwm/

Bibliographie

[00000] [1] J. Brossard, Comparaison des “normes” $L^{p}$ du processus croissant et de la variable maximale pour les martingales régulières à deux indices. Théorème local correspondant, Ann. Probab. 8 (1980), 1183-1188. | Zbl 0454.60047

[00001] [2] J. Brossard, Régularité des martingales à deux indices et inégalités de normes, in: Processus Aléatoires à Deux Indices, Lecture Notes in Math. 863, Springer, Berlin, 1981, 91-121. | Zbl 0473.60045

[00002] [3] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42. | Zbl 0301.60035

[00003] [4] R. Cairoli, Une inégalité pour martingales à indices multiples et ses applications, in: Séminaire de Probabilités V, Lecture Notes in Math. 124, Springer, Berlin, (1970), 1-27. | Zbl 0218.60045

[00004] [5] J. A. Chao, Hardy spaces on regular martingales, in: Martingale Theory in Harmonic Analysis and Banach Spaces, Lecture Notes in Math. 939, Springer, Berlin, 1982, 18-28.

[00005] [6] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. | Zbl 0358.30023

[00006] [7] S. Fridli and P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system, Acta Math. Hungar. 45 (1985), 223-234. | Zbl 0577.42021

[00007] [8] A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Notes Series, Benjamin, New York, 1973.

[00008] [9] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), 3-9. | Zbl 0001.13504

[00009] [10] N. R. Ladhawala, Absolute summability of Walsh-Fourier series, Pacific J. Math. 65 (1976), 103-108. | Zbl 0318.42023

[00010] [11] C. Métraux, Quelques inégalités pour martingales à paramètre bidimensionnel, in: Séminaire de Probabilités XII, Lecture Notes in Math. 649, Springer, Berlin, 1978, 170-179. | Zbl 0381.60042

[00011] [12] F. Móricz, On double cosine, sine and Walsh series with monotone coefficients, Proc. Amer. Math. Soc. 109 (1990), 417-425. | Zbl 0741.42010

[00012] [13] F. Móricz, On Walsh series with coefficients tending monotonically to zero, Acta Math. Acad. Sci. Hungar. 38 (1981), 183-189. | Zbl 0479.42020

[00013] [14] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990. | Zbl 0727.42017

[00014] [15] P. Simon, Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 28 (1985), 87-101. | Zbl 0586.43001

[00015] [16] P. Simon and F. Weisz, Hardy-Littlewood type inequalities for Vilenkin-Fourier coefficients, Anal. Math., to appear. | Zbl 0913.42020

[00016] [17] N. Y. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 363-400 (in Russian); English transl.: Amer. Math. Soc. Transl. 28 (1963), 1-35.

[00017] [18] F. Weisz, Hardy spaces and Cesàro means of two-dimensional Fourier series, in: Approximation Theory and Function Series (Budapest, 1995), Bolyai Soc. Math. Stud. 5, Budapest, 1996, 353-367. | Zbl 0866.42019

[00018] [19] F. Weisz, Inequalities relative to two-parameter Vilenkin-Fourier coefficients, Studia Math. 99 (1991), 221-233. | Zbl 0728.60046

[00019] [20] F. Weisz, Martingale Hardy Spaces and their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. | Zbl 0796.60049

[00020] [21] F. Weisz, Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series, Studia Math. 117 (1996), 173-194. | Zbl 0839.42009

[00021] [22] F. Weisz, Two-parameter Hardy-Littlewood inequalities, ibid. 118 (1996), 175-184. | Zbl 0864.42003