The Molecular Characterization of Weighted Hardy Spaces

We first define molecules for weighted Hardy spaces and prove their molecular characters. As an application, we give sufficient conditions on the kernel k such that the convolution operator Tf=k*f is bounded on weighted Hardy spaces Hpw(Rn), w∈A1. We also get the Hpw(R), 1/2<p⩽1, boundedness of the Hilbert transform and the Hpw(Rn), n/(n+1)<p⩽1, boundedness of the Riesz transforms.