If $x$ is a bounded function on $\mathbf{Z}$, the multiplier with symbol $x$ (denoted by $M_x)$ is defined by $(M_{x}f)\widehat{}=x\widehat{f}$, $f\in L^2\left(\mathbf{T}\right)$. We give some conditions on $x$ ensuring the “interpolation inequality” $\parallel M_{x}f{\parallel }_{L^p}\le C\parallel f{\parallel }_{L^1}^{\alpha }\parallel M_{x}f{\parallel }_{L^q}^{1-\alpha }$ (here $1\<p\<q$ and $\alpha =\alpha (p,q,x)$ is between 0 and 1). In most cases considered $M_x$ fails to have stronger $L^1$-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E\subset \mathbf{Z}$ every positive sequence in ${\ell }^2\left(E\right)$ can be majorized by the sequence $\{$$...$

Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the $L^p$-situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.