\(L^1\rightarrow L^q\) Poincaré inequalities for \(0<q<1\) imply representation formulas
The authors shows that an \(L^1\) to \(L^q\) Poincaré inequality for some \(0<q<1\), i.e., \[ \left(\not \int_B|f-f_B |^q d\nu \right)^{1/q}\leq Cr(B) \not \int_B gd\mu, \] for all metrie balls \(B \subset B_0\), will suffice to imply the representation formula. As an immediate corollary, the authors can show that the weak-type condition \[ \sup_{ \lambda>0} {\lambda\nu \Bigl(\biggl\{x\in B:\bigl|f(x)-f_B\bigr|>\lambda \biggr\} \Bigr) \over\nu(B)}\leq Cr(B) \not \int_B gd\mu, \] also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved.
scientific article; zbMATH DE number 1757441
scientific article