Essential norms of weighted composition operators between Hardy spaces $H^p$ and $H^q$ for $1\leq p,q \leq \infty$

Abstract: We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^p$ and $H^q$ for $1\leq p,q\leq\infty.$ In particular we give some estimates for the cases $1=p\leq q\leq\infty$ and $1\leq q<p\leq\infty.$