  Following a suggestion of Hovey and Strickland, we study the category of
$K(k) \vee K(k+1) \vee \cdots \vee K(n)$-local spectra. When $k = 0$, this is
equivalent to the category of $E(n)$-local spectra, while for $k = n$, this is
the category of $K(n)$-local spectra, both of which have been studied in detail
by Hovey and Strickland. Based on their ideas, we classify the localizing and
colocalizing subcategories, and give characterizations of compact and
dualizable objects. We construct an Adams type spectral sequence and show that
when $p \gg n$ it collapses with a horizontal vanishing line above filtration
degree $n^2+n-k$ at the $E_2$-page for the sphere spectrum. We then study the
Picard group of $K(k) \vee K(k+1) \vee \cdots \vee K(n)$-local spectra, showing
that this group is algebraic, in a suitable sense, when $p \gg n$. We also
consider a version of Gross--Hopkins duality in this category. A key concept
throughout is the use of descent.
