Arithmetic of certain $\ell$-extensions ramified at three places

Let $\ell$ be a regular odd prime, $k$ the $\ell$ th cyclotomic field and $K= k(\sqrt[\ell]{a})$, where $a$ is a positive integer such that there are exactly three places not over $\ell$ that ramify in $K_\infty/k_\infty$. Here $K_\infty$ (resp. $k_\infty$) denotes the cyclotomic $\mathbb Z_\ell$-extension of $K$ (resp. of $k$).
Under assumption that $a$ has exactly three prime divisors $p_1,p_2, p_3$ we study the $\ell$-class group ${\rm Cl}_\ell(K)$ of $K$ and the Iwasawa module (the Tate module) of $K$.
We prove that for $\ell>3$ the order of ${\rm Cl}_\ell(K)$, which we denote by $\ell^r$ satisfies the condition $r\geqslant 2$. Calculation of cohomologies of some groups of unit yields that $r\geqslant \ell-1$ or $r$ is odd. This implies $r>2$ for $\ell>3$. If $\ell=3$ we describe the structure of the Tate module $T_\ell(K_\infty)$.
Some of these results are generalized to the case, when $a$ has two natural divisors $p$ and $q$ such that $q$ remains prime in $K_\infty$ and $p$ splits into two factors.