Corresponding to any (m−1)-tuple of semi-spectral measures on the unit circle, a weighted Dirichlet-type space is introduced and studied. We find that every analytic m-isometry which satisfies a certain set of operator inequalities can be represented as the operator of multiplication by the coordinate function on such a weighted Dirichlet-type space. This extends a result of Richter as well as of Olofsson on analytic 2-isometries. We also prove that all left invertible m-concave operators satisfying the aforementioned operator inequalities admit a Wold-type decomposition. This generalizes a result of Shimorin on a class of 3-concave operators.