Regularity for multi-phase problems at nearly linear growth

Minima of the log-multiphase variational integral w ↦ ∫ Ω [ | D w | log ⁡ ( 1 + | D w | ) + a ( x ) | D w | q + b ( x ) | D w | s ] d x , have locally Hölder continuous gradient under sharp quantitative bounds linking the growth powers ( q , s ) to the Hölder exponents of the modulating coefficients a ( ⋅ ) and b ( ⋅ ) respectively.