We resolve a long-standing question on completeness of the nonorthogonal Mexican hat wavelet system, in L for 1 < p < 2 and in the Hardy space H for 2/3 < p ≤ 1. Tools include the discrete Calderón condition, a generalization of the Daubechies frame criterion to a weighted L space, and imbeddings of that weighted space into L and Hardy spaces.

In the present work the space  $L_{p;r} $ which is continuously embedded into $L_{p} $  is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is...

Abstract We provide an improvement of Calderón and Torchinsky’s version [ 5] the Hörmander multiplier theorem on Hardy spaces $H^p$ ($0&amp;lt;p&amp;lt;\infty $), substituting Sobolev space $L_s^2(A_0)$ by Lorentz–Sobolev $L_s^{\tau ^{(s,p)},\min (1,p) }(A_0)$, where $\tau ^{(s,p)} =\frac{n}{s-(n/\min{(1,p)}-n)}$ $A_0$ is annulus $\{\xi \in{\mathbb{R}}^n:\,\, 1/2&amp;lt;|\xi |&amp;lt;2\}$. Our ...