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Search: MSC category 42B35 ( Function spaces arising in harmonic analysis )

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1. CJM Online first

Cao, Jun; Jiang, Renjin; Yang, Dachun; Zhang, Junqiang
Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$.

Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform
Categories:42B30, 42B35, 35J70

2. CJM 2013 (vol 66 pp. 1382)

Wu, Xinfeng
Weighted Carleson Measure Spaces Associated with Different Homogeneities
In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein.

Keywords:composition of operators, weighted Carleson measure spaces, duality
Categories:42B20, 42B35

3. CJM 2012 (vol 65 pp. 510)

Blasco de la Cruz, Oscar; Villarroya Alvarez, Paco
Transference of vector-valued multipliers on weighted $L^p$-spaces
We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be below one. We also develop some ad-hoc methods which apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allow us to extend results to transference of maximal multipliers and provide transference of Littlewood-Paley inequalities.

Keywords:Fourier multipliers, restriction theorems, weighted spaces
Categories:42B15, 42B35

4. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
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
Categories:42B35, 42C10, 30D50, 28A80

5. CJM 2006 (vol 58 pp. 1121)

Bownik, Marcin; Speegle, Darrin
The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates
The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.

Keywords:frame, Riesz basic sequence, wavelet, Gabor system, frame of translates, paving conjecture
Categories:42B25, 42B35, 42C40

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