We call a \(C^2\) function \(u:\Omega \to \mathbb R\) \(m\)-subharmonic (\(m\)-sh) if \(\Omega \subset \mathbb C ^n\) and the forms \[ (dd^c u)^k \wedge \beta ^{n-k} \] are positive for \(k=1, \dots , m\) with \(\beta = dd^c ||z||^2\). If \(u\) is subharmonic but not smooth then one can define an \(m\)-sh function via inequalities for currents. A domain is strictly \(m\)-pseudoconvex if it has a smooth \(m\)-subharmonic defining function. The author considers the Dirichlet problem for the Hessian equation \[ (dd^c u)^m \wedge \beta ^{n-m} = f\beta ^n , \] for given nonnegative \(f\in L^p (\Omega )\) with \(mp>n\) and \(C^{1,1}\) boundary data. He proves that the solution \(u\) is Hölder continuous, with an explicit exponent, if \(f\) is bounded near the boundary or has a controlled growth there.