A prosthetic valve structure, for use at a native atrioventricular heart valve, includes a frame, prosthetic leaflets, an upper skirt, and a lower skirt. The frame circumscribes a longitudinal axis of the structure. The leaflets are configured to facilitate upstream-to-downstream bloodflow through the structure. The upper skirt has an upstream perimeter. The lower skirt is downstream of the upper skirt, and has a downstream perimeter. The upper skirt extends away from the lower skirt, in an upstream direction and radially outward toward the upstream perimeter. The lower skirt extends away from the upper skirt, in a downstream direction and radially outward toward the downstream perimeter. A downstream portion of the frame is disposed downstream of at least part of the downstream perimeter of the lower skirt.
