Embedding Theorems for $\mathbf{P}$-nary Hardy and $VMO$ Spaces

In the present paper several embedding theorems of P. L. Ul'yanov type for Hölder spaces connected with $\mathbf{P}$-nary Hardy, $VMO$, $L^1$ and uniform metric on Vilenkin groups are proved. Its sharpness is also established. The sufficient conditions for the convergence of Fourier series with respect to multiplicative systems in Hardy space and uniform metric are also given.