( H p , L p ) -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz

Studia Mathematica (1996)

  • Volume: 120, Issue: 3, page 271-288
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Weisz, Ferenc. "$(H_p,L_p)$-type inequalities for the two-dimensional dyadic derivative." Studia Mathematica 120.3 (1996): 271-288. <http://eudml.org/doc/216337>.

@article{Weisz1996,
abstract = {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.},
author = {Weisz, Ferenc},
journal = {Studia Mathematica},
keywords = {Hardy spaces; p-atom; interpolation; Walsh functions; dyadic derivative; multiple Walsh series; dyadic antiderivative; dyadic finite differences; maximal operator; Hardy-Lorentz spaces},
language = {eng},
number = {3},
pages = {271-288},
title = {$(H_p,L_p)$-type inequalities for the two-dimensional dyadic derivative},
url = {http://eudml.org/doc/216337},
volume = {120},
year = {1996},
}

TY - JOUR
AU - Weisz, Ferenc
TI - $(H_p,L_p)$-type inequalities for the two-dimensional dyadic derivative
JO - Studia Mathematica
PY - 1996
VL - 120
IS - 3
SP - 271
EP - 288
AB - 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.
LA - eng
KW - Hardy spaces; p-atom; interpolation; Walsh functions; dyadic derivative; multiple Walsh series; dyadic antiderivative; dyadic finite differences; maximal operator; Hardy-Lorentz spaces
UR - http://eudml.org/doc/216337
ER -

References

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  12. [12] F. Schipp and W. R. Wade, A fundamental theorem of dyadic calculus for the unit square, Appl. Anal. 34 (1989), 203-218. Zbl0727.42020
  13. [13] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990. Zbl0727.42017
  14. [14] F. Weisz, Cesàro summability of two-dimensional Walsh-Fourier series, Trans. Amer. Math. Soc. (1996), to appear. Zbl0866.42020
  15. [15] F. Weisz, Martingale Hardy spaces and the dyadic derivative, Anal. Math., to appear. Zbl0914.42020
  16. [16] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. Zbl0796.60049
  17. [17] F. Weisz, Some maximal inequalities with respect to two-dimensional dyadic derivative and Cesàro summability, Appl. Anal., to appear. Zbl0861.42021
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