## 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

### MSC:

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

### Keywords:

reverse Hölder inequalities