Multipliers in the Hardy spaces $H_p(D^m)$ with $p\in (0,1]$ and approximation properties of summability methods for power series

For $p\in (0,1]$ conditions on a number sequence $\{\lambda _k\}_0^\infty$ are indicated ensuring that the multiplier operator $\sum _{k=0}^\infty c_k z^k \mapsto \sum _{k=0}^\infty \lambda _k c_k z^k$ is continuous in the Hardy space $H_p(D)$ (here $D$ can also be a polydisc $D^m$). Some sufficient conditions are also established. These results are used to find out the precise order of approximation of multiple power series by Bochner–Riesz means and to evaluate the $K$-functional for a pair of spaces related to the polyharmonic operator.