As shown by the author in
{\em Proc.\ Amer.\ Math.\ Soc.} 115 (1992) 345–352, for every metric space
$(K,d)$ with compact closed balls one has $(\mbox{lip}\,\varphi(K))^{**} =
\mbox{Lip} \,\varphi (K),$ where $\varphi$ is any majorant (that is,
non-decreasing function on ${\Bbb R}_+$ with $\varphi (0+) = \varphi (0) = 0$)
such that $\varphi(t)/t$ monotonically tends to $+\infty$ as $t\rightarrow 0.$
Here $\mbox{Lip}\,\varphi(K)$ is the Lipschitz space on $K$ with respect to the
metric $\varphi(d),\ \mbox{lip}\,\varphi(K)$ is the corresponding
’little‘ Lipschitz space of functions vanishing ’at
infinity‘, and ’=‘ means ’canonically isometrically
isomorphic‘. The main idea of the proof consisted of finding a normed space
$M$ such that $M^* = \mbox{Lip}\,\varphi(K)$ and $M^c =
(\mbox{lip}\,\varphi(K))^*,$ where ’$c$‘ stands for the completion,
and identifying $M$ with the space of Borel measures on $K$ equipped with the
Kantorovich norm. In the present paper, this argument is carried over to
generalized Lipschitz spaces on ${\Bbb R}^n$ defined in terms of higher order
differences. For an integer $k$ and for a majorant $\varphi$ with $\lim_{t
\rightarrow 0}\varphi(t)/t ^k = +\infty,$ define $\Lambda ^k_\varphi$ to be the
space of all bounded functions $f$ on ${\Bbb R}^n$ such that for some constant
$C,\ \omega_k(f\,;t) \leq C\varphi(t)$ for all $t\geq 0$, where
$\omega_k(f\,;\cdot)$ is the $k$th modulus of continuity of $f$. Let $\lambda
^k_\varphi$ be the (closed linear separable) subspace in $\Lambda ^k_\varphi$
which consists of functions $f$ vanishing at ’infinity' and such that
$\lim_{t \rightarrow 0}\omega _k(f\,; t)/\varphi (t) = 0.$ We introduce an
appropriate analogue $\Vert \cdot \Vert_{k,\varphi}$ of the Kantorovich norm on
the space $M$ of finite Borel measures on ${\Bbb R}^n$ with compact support. The
properties of this norm for $k \geq 2$ are significantly different from those of
the Kantorovich norm ($k = 1$) which reflects the difference in analytic nature
of generalized Lipschitz spaces versus classical ones. However, the core of the
duality theory survives, and it is shown that $(M, \Vert\cdot \Vert _{k,
\varphi})^* = \Lambda ^k _\varphi,\ \ (M, \Vert \cdot \Vert _{k,\varphi})^c =
(\lambda ^k _\varphi)^*$ and consequently, $(\lambda ^k _\varphi )^{**} = \Lambda
^k _\varphi$. Several applications of these results are discussed, and a few open
problems are formulated.
1991 Mathematics Subject Classification:
46E15, 46E35.
