Self-improvement of Poincaré type inequalities associated with approximations of the identity and semigroups
The authors develop a general theory to obtain self-improving properties from a starting Poincaré type inequality of the form \[ \frac1{|Q|}\int_Q|f(x)-S_{t_Q}f(x)|\,dx\leq a(Q,f),\tag{1} \] where the functional \(a\) satisfies the \(D_r\) condition: for each cube \(Q\) and any family \(\{Q_i\}_i\) of pairwise disjoint subcubes of \(Q\), \[ \sum_ia(Q_i)^r|Q_i|\leq C_a^ra(Q)^r|Q|, \] and where the family of linear operators \(\{S_t\}_{t>0}\) commutes and admits a representation \[ S_ff(x)=\int_{\mathbb R^n}s_t(x,y)f(y)\,dy, \] with \(s_t\) having suitable decay properties. The proofs are based on the Whitney covering lemma and good-\(\lambda\) inequalities. One of the main results is the following \(L^{r,\infty}\) self-improvement property: if (1) holds, then \[ \| f-S_{t_Q}f\|_{L^{r,\infty},Q}\leq Ca(Q). \] Extensions to the case of \(A_\infty\) weights are also considered. As an application, the authors study different examples involving fractional averages, multiparameter generalized Hardy inequalities, or weaker norms in terms of Orlicz spaces.
scientific article; zbMATH DE number 6160847
scientific article