Parabolic function spaces with mixed norm

Abstract:

The spaces $\mathcal {H}_\alpha ^p$ of parabolic Bessel potentials were introduced by B. F. Jones and R. J. Bagby. We prove a Sobolev-type imbedding theorem for $\mathcal {H}_\alpha ^{{p_1},{p_2}}$ (multinormed versions of $\mathcal {H}_\alpha ^p$) when $\alpha$ is a positive integer k, $1 < {p_1}$, ${p_2} < \infty$. In particular this theorem holds for $W_{2l,l}^p$, since $\mathcal {H}_{2l}^p \equiv W_{2l,l}^p$. We use the concepts of parabolic Riesz transforms and half-time derivatives introduced by us elsewhere.