We characterize H¨older continuously differentiable m dimensional submanifolds of Euclidean space among m rectifiable sets S in terms of growth conditions on the m density ratios of the Hausdorff measure Hm S.

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It is proved that the natural embedding of a separable Banach space$X$into the corresponding Bourgain–Pisier space extends $\mathcal{L}_{\infty}$ -valued operators.

Let $\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}$ be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in $\mathbb{R}^{n}$ . We show that the$L$^{$p$}norm, 1<$p$<∞, of the related maximal operator $$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$ is at most$C$(log($N$+2))^{$n$/2}. We show that this bound is sharp.