A wall structure for bordering a combustion chamber of a burner has a top plate that is exposed to the combustion chamber and a bottom plate in contact with the top plate in contact zones on a side facing away from the combustion chamber. The top plate and/or the bottom plate are shaped so that a channel system is created between the top plate and the bottom plate. The top plate contains first openings which communicate with the channel system. In addition, the top plate and the bottom plate have joint second openings in the contact zones communicating with a feed space which is arranged on a side of the bottom plate facing away from the combustion chamber.