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1996 | 120 | 3 | 271-288
Tytuł artykułu

$(H_p,L_p)$-type inequalities for the two-dimensional dyadic derivative

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It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p,q}$ to $L_{p,q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_1,L_1)$. As a consequence we show that the dyadic integral of a ∞ function $f ∈ L_1$ is dyadically differentiable and its derivative is f a.e.
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autor
  • Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary , weisz@ludens.elte.hu
Bibliografia
  • [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988.
  • [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.
  • [3] P. L. Butzer and W. Engels, Dyadic calculus and sampling theorems for functions with multidimensional domain, Inform. and Control 52 (1982), 333-351.
  • [4] P. L. Butzer and H. J. Wagner, On dyadic analysis based on the pointwise dyadic derivative, Anal. Math. 1 (1975), 171-196.
  • [5] P. L. Butzer and H. J. Wagner, Walsh series and the concept of a derivative, Appl. Anal. 3 (1973), 29-46.
  • [6] A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Notes Ser., Benjamin, New York, 1973.
  • [7] Gy. Gát, On the two-dimensional pointwise dyadic calculus, J. Approx. Theory, to appear.
  • [8] J. Neveu, Discrete-Parameter Martingales, North-Holland, 1971.
  • [9] F. Schipp, Über einen Ableitungsbegriff von P. L. Butzer und H. J. Wagner, Math. Balkanica 4 (1974), 541-546.
  • [10] F. Schipp, Über gewissen Maximaloperatoren, Ann. Univ. Sci. Budapest. Sect. Math. 18 (1975), 189-195.
  • [11] F. Schipp and P. Simon, On some $(H,L_1)$-type maximal inequalities with respect to the Walsh-Paley system, in: Functions, Series, Operators, Budapest 1980, Colloq. Math. Soc. János Bolyai 35, North-Holland, Amsterdam, 1981, 1039-1045.
  • [12] F. Schipp and W. R. Wade, A fundamental theorem of dyadic calculus for the unit square, Appl. Anal. 34 (1989), 203-218.
  • [13] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  • [14] F. Weisz, Cesàro summability of two-dimensional Walsh-Fourier series, Trans. Amer. Math. Soc. (1996), to appear.
  • [15] F. Weisz, Martingale Hardy spaces and the dyadic derivative, Anal. Math., to appear.
  • [16] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [17] F. Weisz, Some maximal inequalities with respect to two-dimensional dyadic derivative and Cesàro summability, Appl. Anal., to appear.
  • [18] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.
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