# Two-parameter Hardy-Littlewood inequality and its variants

Studia Mathematica (2000)

- Volume: 139, Issue: 1, page 9-27
- ISSN: 0039-3223

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topChen, Chang-Pao, and Luor, Dah-Chin. "Two-parameter Hardy-Littlewood inequality and its variants." Studia Mathematica 139.1 (2000): 9-27. <http://eudml.org/doc/216714>.

@article{Chen2000,

abstract = {Let s* denote the maximal function associated with the rectangular partial sums $s_\{mn\}(x,y)$ of a given double function series with coefficients $c_\{jk\}$. The following generalized Hardy-Littlewood inequality is investigated: $||s*||_\{p,μ\}≤C_\{p,α,β\} \{Σ_\{j=0\}^∞Σ_\{k=0\}^∞(j̅ )^\{p-α-2\}(k̅)^\{p-β-2\}|c_\{jk\}|^p \}^\{1/p\}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_\{jk\}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_\{mn\}(x,y)$ is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].},

author = {Chen, Chang-Pao, Luor, Dah-Chin},

journal = {Studia Mathematica},

keywords = {Hardy-Littlewood inequality; trigonometric system; Walsh system; Legendre system; maximal function},

language = {eng},

number = {1},

pages = {9-27},

title = {Two-parameter Hardy-Littlewood inequality and its variants},

url = {http://eudml.org/doc/216714},

volume = {139},

year = {2000},

}

TY - JOUR

AU - Chen, Chang-Pao

AU - Luor, Dah-Chin

TI - Two-parameter Hardy-Littlewood inequality and its variants

JO - Studia Mathematica

PY - 2000

VL - 139

IS - 1

SP - 9

EP - 27

AB - Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: $||s*||_{p,μ}≤C_{p,α,β} {Σ_{j=0}^∞Σ_{k=0}^∞(j̅ )^{p-α-2}(k̅)^{p-β-2}|c_{jk}|^p }^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_{mn}(x,y)$ is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].

LA - eng

KW - Hardy-Littlewood inequality; trigonometric system; Walsh system; Legendre system; maximal function

UR - http://eudml.org/doc/216714

ER -

## References

top- [1] R. Askey and S. Wainger, Integrability theorems for Fourier series, Duke Math. J. 33 (1966), 223-228. Zbl0136.36501
- [2] L. A. Balashov, Series with respect to the Walsh system with monotone coefficients, Sibirsk. Mat. Zh. 12 (1971), 25-39 (in Russian). Zbl0224.42010
- [3] R. P. Boas, Integrability Theorems for Trigonometric Transforms, Springer, Berlin, 1967. Zbl0145.06804
- [4] T. W. Chaundy and A. E. Jolliffe, The uniform convergence of a certain class of trigonometric series, Proc. London Math. Soc. (2) 15 (1916), 214-216.
- [5] C.-P. Chen, Integrability and L-convergence of multiple trigonometric series, Bull. Austral. Math. Soc. 49 (1994), 333-339. Zbl0795.42007
- [6] C.-P. Chen, Weighted integrability and ${L}^{1}$-convergence of multiple trigonometric series, Studia Math. 108 (1994), 177-190. Zbl0821.42007
- [7] C.-P. Chen and G.-B. Chen, Uniform convergence of double trigonometric series, ibid. 118 (1996), 245-259.
- [8] Y.-M. Chen, On the integrability of functions defined by trigonometric series, Math. Z. 66 (1956), 9-12. Zbl0071.06002
- [9] Y.-M. Chen, Some asymptotic properties of Fourier constants and integrability theorems, ibid. 68 (1957), 227-244. Zbl0078.05503
- [10] M. I. D'yachenko, On the convergence of double trigonometric series and Fourier series with monotone coefficients, Math. USSR-Sb. 57 (1987), 57-75. Zbl0654.42017
- [11] 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. Zbl0577.42021
- [12] A. E. Jolliffe, On certain trigonometric series which have a necessary and sufficient condition for uniform convergence, Proc. Cambridge Philos. Soc. 19 (1921), 191-195.
- [13] L. Leindler, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math. (Szeged) 31 (1970), 279-285. Zbl0203.06103
- [14] L. Leindler, Some inequalities of Hardy-Littlewood type, Anal. Math. 20 (1994), 95-106. Zbl0816.42017
- [15] M. M. H. Marzug, Integrability theorem of multiple trigonometric series, J. Math. Anal. Appl. 157 (1991), 337-345.
- [16] F. Móricz, On Walsh series with coefficients tending monotonically to zero, Acta Math. Acad. Sci. Hungar. 38 (1981), 183-189. Zbl0479.42020
- [17] F. Móricz, On the maximum of the rectangular partial sums of double trigonometric series with nonnegative coefficients, Anal. Math. 15 (1989), 283-290. Zbl0756.42010
- [18] F. Móricz, On double cosine, sine, and Walsh series with monotone coefficients, Proc. Amer. Math. Soc. 109 (1990), 417-425 Zbl0741.42010
- [19] F. Móricz, On the integrability and ${L}^{1}$-convergence of double trigonometric series, Studia Math. 98 (1991), 203-225. Zbl0724.42015
- [20] F. Móricz, F. Schipp and W. R. Wade, On the integrability of double Walsh series with special coefficients, Michigan Math. J. 37 (1990), 191-201. Zbl0714.42017
- [21] J. R. Nurcombe, On the uniform convergence of sine series with quasimonotone coefficients, J. Math. Anal. Appl. 166 (1992), 577-581. Zbl0756.42006
- [22] B. Ram and S. S. Bhatia, On weighted integrability of double cosine series, ibid. 208 (1997), 510-519. Zbl0879.42008
- [23] F. Schipp, P. Simon and W. R. Wade, Walsh Series, An Introduction to Dyadic Harmonic Analysis, IOP Publishing and Akadémiai Kiadó, Budapest, 1990.
- [24] S. B. Stechkin, On power series and trigonometric series with monotone coefficients, Uspekhi Mat. Nauk 18 (1963), no. 1, 173-180 (in Russian). Zbl0113.27502
- [25] G. Szegő, Orthogonal Polynomials, 4th ed., Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.
- [26] F. Weisz, Inequalities relative to two-parameter Vilenkin-Fourier coefficients, Studia Math. 99 (1991), 221-233. Zbl0728.60046
- [27] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. Zbl0796.60049
- [28] F. Weisz, Two-parameter Hardy-Littlewood inequalities, Studia Math. 118 (1996), 175-184. Zbl0864.42003
- [29] T. F. Xie and S. P. Zhou, The uniform convergence of certain trigonometric series, J. Math. Anal. Appl. 181 (1994), 171-180. Zbl0791.42004
- [30] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. 12 (1913), 41-70. Zbl44.0300.03
- [31] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1968. Zbl0157.38204

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