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A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$

  Published:2010-07-06
 Printed: Oct 2010
  • Hong Yue,
    Trine University, Angola, IN, U.S.A.
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Abstract

A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$.
Keywords: Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension
MSC Classifications: 42B35, 42C10, 30D50, 28A80 show english descriptions Function spaces arising in harmonic analysis
Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part
Fractals [See also 37Fxx]
42B35 - Function spaces arising in harmonic analysis
42C10 - Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
30D50 - Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part
28A80 - Fractals [See also 37Fxx]
 

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