Fourier transform of anisotropic Hardy spaces

Abstract:

We show that if $f$ is in an anisotropic Hardy space $H_A^p$, $0 < p \leq 1$, with respect to a dilation matrix $A$, then its Fourier transform $\hat {f}$ satisfies the pointwise estimate \[ |\hat f(\xi )| \le C ||f||_{H^p_A} \rho _*(\xi )^{\frac {1}{p}-1}.\] Here, $\rho _*$ is a quasi-norm associated with the transposed matrix $A^*$. This leads to necessary conditions for functions $m$ to be multipliers on $H_A^p$, as well as further pointwise characterizations on $\hat {f}$ and a generalization of the Hardy-Littlewood inequality on the integrability of $\hat {f}$. This last result is strengthened through the use of rearrangement functions.