The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f ∈ L 2, − µ (ℝ 3 ), wherewhere x = ( x 1 , x 2 , x 3 ) is the Cartesian system in ℝ 3 , x ′ = ( x 1 , x 2 ), , µ∈ℝ + \ℤ. We prove the decomposition f = − ∇ u + g , with g divergence free and u is a solution to the problem in ℝ 3Given f ∈ L 2, − µ (ℝ 3 ) we show the existence of u ∈ H   1 −µ (ℝ 3 ) such thatwhereSince f, u, g are defined in ℝ 3 we need a sufficiently fast decay of these functions as | x |→∞. Copyright © 2010 John Wiley & Sons, Ltd.