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


Published:20100706
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
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine 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 $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.
Keywords: 
Haar wavelets, Q spaces, JohnNirenberg inequality, Greedy expansion, selfaffine, fractal, Box dimension
Haar wavelets, Q spaces, JohnNirenberg inequality, Greedy expansion, selfaffine, 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]
