Let Ω ⋐ ℂ n be a bounded strongly m -pseudoconvex domain ( 1 ≤ m ≤ n ) and μ a positive Borel measure with finite mass on Ω . We solve the Hölder continuous subsolution problem for the complex Hessian equation ( d d c u ) m ∧ β n - m = μ on Ω . Namely, we show that this equation admits a unique Hölder continuous solution on Ω ¯ with given Hölder continuous boundary values if it admits a Hölder continuous subsolution on Ω ¯ . The main step in solving the problem is to establish a new capacity estimate showing that the m -Hessian measure of a Hölder continuous m -subharmonic function on Ω ¯ with zero boundary values is dominated by the m -Hessian capacity with respect to Ω with an (explicit) exponent τ > 1 .