Are Morrey spaces reflexive? Since $L^{p,0}=L^p$ and $L^1$ is not reflexive, thus in general Morrey space is not reflexive, but how about for $L^{p,\lambda}$ with $1<p<+\infty$ and $0<\lambda<n$, where $n$ is the dimension of domain. 
What's more, it seems that the dual space for Morrey spaces are not clear so far?

For any function space defined as "some supremum is finite" without further conditions like continuity or vanishing, you should expect that:
(a) the space is nonreflexive and nonseparable;
(b) its dual is too large to be described in concrete terms.
The reason for both things is that $\ell_\infty$ embeds into the space (recall that every subspace of a nonreflexive space must be nonreflexive). In case of Morrey spaces, take a well-separated sequence of balls $B_n$ with quickly decaying radii $r_n$, and define, for any sequence $c\in\ell_\infty$, 
$$f_c = \sum_n c_n r_n^{(\lambda-n)/p}\chi_{B_n}$$ 
This is constructed so that 
$$\|c\|_\infty\le \|f\|_{\lambda,p}\le K\|c\|_\infty$$
for some constant $K$.
