On the theory of homogeneous Lipschitz spaces and Campanato spaces (Q788240)

Let k be a positive integer, let \(P_{k-1}\) be the set of polynomials on \({\mathbb{R}}^ n\) of degree not exceeding k-1, and let \({\mathcal S}'\) be the set of tempered distributions on \({\mathbb{R}}^ n\). For \(0<\alpha<k\), let \(\Lambda_{\alpha,k}\) be the space of equivalence classes [f] in \({\mathcal S}'\) modulo \(P_{k-1}\) such that \[ \| f\|_{\alpha,k}=\sum_{| \nu | =k}\sup_{t\in {\mathbb{R}}^+}\sup_{x\in {\mathbb{R}}^ n}t^{(k-\alpha)/2}| D^{\nu}f(x,t)|<\infty, \] where f(x,t) is the Gauss-Weierstrass integral of f and D denotes (partial) differentiation with respect to \(x_ 1,...x_ n\). Consider also the space \(L(\alpha\),p,k-1) of equivalence classes modulo \(P_{k-1}\) of locally integrable functions f for which \[ \| f\|_{L(\alpha,p,k-1)}=\sup_{Q\subset {\mathbb{R}}^ n}| Q|^{-\alpha /n}[\frac{1}{| Q|}\int_{Q}| f(x)- P_ Qf(x)|^ Pdx]^{1/p}<\infty, \] where Q is a ball and \(P_ Qf\) is the unique element of \(P_{k-1}\) such that \(\int_{Q}[f(x)-P_ qf(x)]x^{\nu}dx=0\) for 0\(\leq | \nu | \leq k-1\). Such spaces occur in the duality theory of multi-dimensional Hardy spaces as discussed by \textit{M. Taibleson} and \textit{G. Weiss} [Astérisque 77, 67- 151 (1980; Zbl 0472.46041)]. The main result of the present paper asserts that the spaces \(\Lambda_{\alpha,k}\) and \(L(\alpha\),p,k-1) coincide and that their norms are equivalent. An earlier result along related lines was obtained by \textit{B. Grevholm} for p in the range \(1\leq p<\infty\) using interpolation theory [Math. Scand. 26, 241-254 (1971; Zbl 0212.460)]. The result here is valid for 1\(\leq p\leq \infty\) and is proved by elementary methods.