Growth envelopes in Muckenhoupt weighted function spaces: the general case
Let \(X_w\) be either \(B^s_{p,q} (\mathbb R^n,w)\) or \(F^s_{p,q} (\mathbb R^n, w)\) with \(0<p<\infty\), \(0<q \leq \infty\) and \[ n \max \big( \tfrac{1}{p} -1, 0 \big) 0, \] is the growth envelope function, where \(f^*\) is the usual rearrangement of \(f\). Let \(\mu\) be the Borel measure generated by the distribution function \(- \log E^{X_w}_G (t)\), \(0<t\leq \epsilon\). Let \(u^{X_w}_G\) be the infimum of all \(0<v \leq \infty\) such that \[ \biggl( \int^\epsilon_0 \biggl( \frac{f^* (t)}{E^{X_w}_G (t) } \biggr)^v \mu (dt) \biggr)^{1/v} \leq c \, \| f \, | X_w \|. \] Then \((E^{X_w}_G (t), u^{X_w}_G )\) is called the growth envelope of \(X_w\). The first main result of the paper (Theorem 3.6) is the assertion that the growth envelope for \(X_w\) with \(w\in A_1\) (Muckenhoupt class) and \(\int_B w(x) \, dx \geq c >0\) for all balls of radius 1 is the same as in the unweighted case \(w=1\). Furthermore, there are detailed, but less complete assertions if \(w \in A_\infty\). The results are illustrated by examples typically with different behaviour of type \(|x|^\alpha \big| \log |x| \, \big|^\beta\) near 0 and \(\infty\).
scientific article; zbMATH DE number 5758464
scientific article