Some new estimates of the Fourier–Bessel transform in the space $\mathbb{L}_2(\mathbb{R}_+)$

The Fourier–Bessel integral transform
$$ g(x)=F[f](x)=\frac1{2^p\Gamma(p+1)}\int_0^{+\infty}t^{2p+1}f(x)j_p(xt)dt $$
is considered in the space $\mathbb{L}_2(\mathbb{R}_+)$. Here, $j_p(u)=((2^p\Gamma(p+1))/(u^p))J_p(u)$ and $J_p(u)$ is a Bessel function of the first kind. New estimates are proved for the integral
$$ \delta^2_N(f)=\int_N^{+\infty}x^{2p+1}g^2(x)dx,\quad N>0, $$
in $\mathbb{L}_2(\mathbb{R}_+)$ for some classes of functions characterized by a generalized modulus of continuity.