  We study Dirichlet forms defined by nonintegrable L\'evy kernels whose
singularity at the origin can be weaker than that of any fractional Laplacian.
We show some properties of the associated Sobolev type spaces in a bounded
domain, such as symmetrization estimates, Hardy inequalities, compact inclusion
in $L^2$ or the inclusion in some Lorentz space. We then apply those properties
to study the associated nonlocal operator $\mathfrak{L}$ and the Dirichlet and
Neumann problems related to the equations $\mathfrak{L}u=f(x)$ and
$\mathfrak{L}u=f(u)$ in $\Omega$.
