p-Summing Multiplication Operators, dyadic Hardy Spaces and atomic Decomposition

We constructively determine the Pietsch measure of the 2-summing multiplication operator $$\mathcal{M}_u:\ell^{\infty} \rightarrow H^p, \quad (\varphi_I) \mapsto \sum \varphi_Ix_Ih_I.$$ Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_u$ involves the Haar coefficients of $u$ and its atomic decomposition.