On Gurov-Reshetnyak classes. (English) Zbl 0713.30018

Let w(x) be a fixed weight function satisfying the strong doubling condition and let \(d\mu =w dx\). The following result is proved. Let \(f:Q_ 0\to {\mathbb{R}}^ m\) \((Q_ 0\) is a cube in \(R^ n)\) satisfy, for some \(\epsilon >0\), some \(q\geq 1\) and for each cube Q dyadic with respect to \(Q_ 0\) the inequality \[ (1)\quad ((average)\quad \int_{Q}| f- f_{Q,w}|^ qd\mu)^{1/q}\leq \epsilon \quad (average)\quad \int_{Q}| f| d\mu, \] where \[ f_{Q,w}=\frac{1}{\mu (Q)}\int_{Q}fd\mu =(average)\quad \int_{0}fd\mu. \] Then for sufficiently small \(\epsilon,\epsilon \leq \epsilon_ 0(n,q)\), there exists a constant \(C_ 0\), depending on n, q and the constants involved in the doubling condition only such that f belongs to \(L^ p(Q_ 0)\) for every p, \(q\leq p<C_ 0/\epsilon\). Moreover, for these values of p, the estimate \[ ((average)\quad \int_{Q}| f-f_{Q,w}|^ pd\mu)^{1/p}\leq C_ 1\quad (average)\quad \int_{Q}| f| d\mu \] holds, with the constant \(C_ 1\) depending on q,n and the constants of the doubling condition only, for each dyadic cube \(Q\subset Q_ 0\). Interesting applications of this result to reverse Hölder inequalities are given. In particular: let \(B^ p_ q(K)\) \((p>q,K>1)\) be the family of all functions satisfying the reverse Hölder inequality \(\| f\|_{Q,q}\leq K\| f\|_{Q,q}\) for each cube Q contained in a fixed cube \(Q_ 0\) in \(R^ n\), and denote by I(K,n,p,q) the optimal integrability exponent for all functions in \(B^ p_ q(K)\); the author gives some information on the asymptotic of I(K,n,p,q) for \(K\to 1\). Finally, a weak form of inequality (1) is investigated (Q is replaced by \(\sigma\) Q, \(0<\sigma <1\), in the left hand side of (1)).
Reviewer: G.Porru


30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations