  Starting from the first renormalized factorization theorem for a process
described at subleading power in soft-collinear effective theory, we discuss
the resummation of Sudakov logarithms for such processes in
renormalization-group improved perturbation theory. Endpoint divergences in
convolution integrals, which arise generically beyond leading power, are
regularized and removed by systematically rearranging the factorization
formula. We study in detail the example of the $b$-quark induced
$h\to\gamma\gamma$ decay of the Higgs boson, for which we resum large
logarithms of the ratio $M_h/m_b$ at next-to-leading logarithmic order. We also
briefly discuss the related $gg\to h$ amplitude.
