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Integrability of double lacunary sine series

Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 110 (1990), 355-364
MSC: Primary 42B05
MathSciNet review: 1021902
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Abstract: We consider mainly the series $ \sum {\sum {{a_{jk}}\sin {2^j}x\sin {2^k}y} } $, which converges to a finite function $ f(x,y)$ a.e. if $ \sum {\sum {a_{jk}^2 < \infty } } $. We show that the (Lebesgue) integrability of $ {x^{ - 1}}{y^{ - 1}}f(x,y)$ over the two-dimensional torus is essentially controlled by the quantity $ \sum\nolimits_{m = 1}^\infty {\sum\nolimits_{n = 1}^\infty {{{\left( {\sum\nol... ...{j = m}^\infty {\sum\nolimits_{k = n}^\infty {a_{jk}^2} } } \right)}^{1/2}}} } $. Our result is an extension of the corresponding one by M. C. Weiss [4] from one-dimensional to two-dimensional lacunary sine series.

References [Enhancements On Off] (What's this?)

  • [1] R. P. Boas, Integrability theorems for trigonometric transforms, Springer-Verlag, Berlin, 1967.
  • [2] F. Móricz, On the $ \left\vert {C,\alpha > 1/2,\beta > 1/2} \right\vert$-summability of double orthogonal series, Acta Sci. Math. (Szeged) 48 (1985), 325-338. MR 810889 (86m:42018)
  • [3] W. Orlicz, Beiträge zur Theorie der Orthogonalentwicklungen, Studia Math. 6 (1936), 20-38.
  • [4] M. C. Weiss, The law of the iterated logarithm for lacunary series and its application to the Hardy-Littlewood series, dissertation, University of Chicago, 1957.
  • [5] A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, 1959. MR 0107776 (21:6498)

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Keywords: Double sine series, Fourier series, lacunarity, a.e. convergence, Lebesgue integrability
Article copyright: © Copyright 1990 American Mathematical Society

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