The exact continuation of a reverse Hölder inequality and Muckenhoupt's conditions
Let \(\varphi\) be a convex downward positive function on \((0,+\infty)\) and for a measurable nonnegative function \(f\) on \([a,b]\), \[ S(\varphi,f, [a,b])= \sup_{a\leq \alpha\leq \beta\leq b}\left(\int^ b_ a \varphi(f)dx\right)\left(\varphi\left(\int^ b_ a f dx\right)\right)^{-1}, \] where \(\int^ b_ a f dx= (b- a)^{- 1}\int^ b_ a f dx\). It is proved that \[ S\bigl(\varphi, f^*,[0,b- a]\bigr)\leq S\bigl(\varphi,f, [a,b]\bigr), \] where \(f^*\) is a nonincreasing rearrangement of \(f\). Several important corollaries of this result are derived, among them the following one. Denote for \(1< q<+\infty\) by \(A_ q(M)\) the class of all functions satisfying Muckenhoupt's \(A_ q\)-condition on \([a,b]\) with constant \(M> 0\). The number \(r= r(q,M)> 1\) is found such that for \(r< p< q\) for some \(N\), \(A_ q(M)\subset A_ p(N)\) and for any \(N\), \(A_ q(M)\not\subset A_ r(N)\).
scientific article; zbMATH DE number 578708