W-SOBOLEV SPACES: HIGHER ORDER AND REGULARITY

Fix a function W(x(1), ... ,x(d)) = Sigma(d)(k=1) W-k(x(k)) where each W-k: R -> R is a right continuous with left limits and strictly increasing function, and consider the W-laplacian given by Delta(W) = Sigma(d)(i=1) partial derivative(xi)partial derivative W-i, which is a generalization of the laplacian operator. In this work we introduce the W-Sobolev spaces of higher order, thus extending the notion of W-Sobolev spaces introduced in Simas and Valentim (2011) [7]. We then provide a characterization of these spaces in terms of a suitable Fourier series, and conclude the paper with some results on elliptic regularity of the problem lambda u - Delta(W)u = f, for lambda >= 0.