arXiv subjects
Publications and source records attributed to Li-An Daniel Wang.
We obtain a differential characterization for the anisotropic Hardy space $H_A^p$ by identifying it with a parabolic Hardy space associated with a general continuous group. This allows $H_A^p$ to be defined using a parabolic differential equation of Calderon and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.
We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb{R}^n) \rightarrow H_A^p (\mathbb{R}^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss.
We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range, and $A_{\infty}$ extrapolation, to the weighted variable Lebesgue spaces. As a consequence we are able to show that a number of different operators from harmonic analysis are bounded on these spaces. The proofs of our extrapolation results are developed in a way that outlines a general approach to proving extrapolation theorems on other Banach function spaces.
We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Stromberg and Torchinsky than the classical atomic decomposition. As an application of the atomic decomposition we show that singular integral operators are bounded on variable Hardy spaces with minimal regularity assumptions on the exponent function.
We extend an estimate of Taibleson and Weiss, regarding Fourier transform of Hardy spaces, to the aniostropic setting. As consequences, we obtain necessary conditions for multiplier operators, and the anisotropic version of the Hardy-Littlewood inequality. This last inequality is strengthened with a rearrangement argument.