Using Fourier series representations of functions on axisymmetric domains, we find weighted Sobolev norms of the Fourier coefficients of a function that yield norms equivalent to the standard Sobolev norms of the function. This characterization is universal in the sense that the equivalence constants are independent of the domain. In particular it is uniform whether the domain contains a part of its axis of rotation or is disjoint from, but maybe arbitrarily close to, the axis. Our characterization using step-weighted norms involving the distance to the axis is different from the one obtained earlier in the book [Bernardi, Dauge, Maday "Spectral methods for axisymmetric domains", Gauthier-Villars, 1999], which involves trace conditions and is domain dependent. We also provide a complement for non cylindrical domains of the proof given in loc. cit. .

翻译：利用轴对称域上函数的傅里叶级数表示，我们找到了函数傅里叶系数的加权Sobolev范数，该范数可产生等价于函数标准Sobolev范数的等价范数。这一表征具有普适性，即等价常数与域无关。特别地，无论域是否包含其旋转轴的一部分，还是与轴不相交但可能任意接近，该结果均保持一致。我们采用涉及到轴距离的阶梯加权范数得到的表征，不同于此前在文献[Bernardi, Dauge, Maday "Spectral methods for axisymmetric domains", Gauthier-Villars, 1999]中获得的需依赖迹条件且与域相关的表征。此外，我们还对上述文献中针对非柱状域给出的证明提供了补充。