The Convergence of Calderón Reproducing Formulae of Two Parameters in $L^p$, in $\mathscr S$ and in $\mathscr S'$

The Calderón reproducing formula is the most important in the study of harmonic analysis, which has the same property as the one of approximate identity in many special function spaces. In this note, we use the idea of separation variables and atomic decomposition to extend single parameter to two-parameters and discuss the convergence of Calderón reproducing formulae of two-parameters in $L^p(\mathbb R^{n_1} \times \mathbb R^{n_2})$, in $\mathscr S(\mathbb R^{n_1} \times \mathbb R^{n_2})$ and in $\mathscr S'(\mathbb R^{n_1} \times \mathbb R^{n_2})$.

ATOMIC DECOMPOSITION OF WEIGHTED TRIEBEL–LIZORKIN SPACES ON SPACES OF HOMOGENEOUS TYPE

We obtain an atomic decomposition for weighted Triebel–Lizorkin spaces on spaces of homogeneous type, using the area function, the discrete Calderón reproducing formula and discrete sequence spaces.

The Calderón Reproducing Formula Associated with the Heisenberg GroupHd

We establish the Calderón reproducing formula for functions inL2on the Heisenberg groupHd. Also, we develop this formula inLp(Hd)with1<p<∞.