Orthogonal decompositions of Sobolev spaces in Clifford analysis

The space L ₂(G;ℂ _m ) of Clifford-algebra-valued functions in bounded domains G of ℝ ^m is decomposed into the orthogonal sum of the subspace of poly-left-monogenic functions of arbitrary order k≥1 and its orthogonal complement and as well into the orthogonal sum of the subspace of polyharmonic functions of arbitrary order k≥1 and its orthogonal complement. The complementary subspaces are given explicitly. In the particular case m=2, complex functions are involved. Although this case has to be treated separately, the results are as before. The proofs are based on proper higher-order Cauchy–Pompeiu formulas and Green functions for powers of the Laplacian. Received: July 4, 2000; in final form: January 7, 2001?Published online: December 19, 2001