Decompositions of the Sobolev Scale and Gradient–Divergence Scale into the Sum of Solenoidal and Potential Subspaces

For the complete Sobolev scale and the gradient–divergence scale, decompositions into direct sums of solenoidal and potential subspaces are found. A smoothing property of solenoidal factorization is proved. Projectors onto the subspaces of solenoidal and potential functions are described.