Reverse Hölder inequalities with constant close to 1
The main result of the paper is: Let \(f\) be a nonnegative function in \(L^ p(\mathbb{R}^ n)\) \((p>1)\) satisfying: \[ \int_ Q\kern-1.25em-f^ pdx\leq(1+\delta)(\int_ Q\kern-1.125em-f dx)^ p,\quad\delta>0\tag{*} \] for every cube \(Q\) with sides parallel to the axes. Then there exist constants \(c\) and \(c'\) depending on \(p\) and \(n\) such that (1) \(f\) belongs to \(L^ r_{loc}\) for every \(r<c/\sqrt\delta\), (2) for fixed \(r\) and every cube \(Q\) it results: \(\int_ Q\kern-1.25em- f^ rdx\leq(1+c'\sqrt\delta)(\int_ Q\kern-1.25em- f dx)^ r\). Results (1) and (2) give precise information on what happens to functions satisfying the reverse Hölder inequality (*) when \(\delta\) tends to zero. The nonincreasing rearrangement method is used for the proof.
scientific article; zbMATH DE number 26525