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In [CGN12], we proved that the renormalized critical Ising magnetization fields $Phi^a:= a^{15/8} sum_{xin a, Z^2} sigma_x , delta_x$ converge as $ato 0$ to a random distribution that we denoted by $Phi^infty$. The purpose of this paper is to establish some fundamental properties satisfied by this $Phi^infty$ and the near-critical fields $Phi^{infty,h}$. More precisely, we obtain the following results. bi [(i)] If $Asubset C$ is a smooth bounded domain and if $m=m_A :=  0$ such that {equation*} log Pb{m u003e x} underset{xto infty}{sim} -c ; x^{16},.{equation*} In particular, this provides an alternative proof that the field $Phi^infty$ is non-Gaussian (another proof of this fact would use the $n$-point correlation functions established in cite{CHI} which do not satisfy Wicku0027s formula). [(ii)] The random variable $m=m_A$ has a smooth {it density} and one has more precisely the following bound on its Fourier transform: $|Eb{e^{i,t m}} |le e^{- tilde{c}, |t|^{16/15}}$. [(iii)] There exists a one-parameter family $Phi^{infty,h}$ of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field. ei

Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits