L 1L q Poincar Inequalities for 0 > q > 1 Imply Representation Formulas

Given two doubling measures and in a metric space (S, ) of homogeneous type, let B 0S be a given ball. It has been a well-known result by now (see [14]) that the validity of an L 1L 1 Poincar inequality of the following form: B|ffB|dvcr(B)Bgd, for all metric balls BB 0S, implies a variant of representation formula of fractional integral type: for -a.e. xB 0, |f(x)fB0|CB0g(y)(x,y)(B(x,(x,y)))d(y)*Cr(B0)(B0)B0g(y)d(y). One of the main results of this paper shows that an L 1 to L q Poincar inequality for some 0 > q > 1, i.e., (B|ffB|qdv)1/qcr(B)Bgd, for all metric balls BB 0, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, sup>0({xB:|f(x)fB|>})(B)Cr(B)Bgd, also implies the same formula. Analogous theorems related to high-order Poincar inequalities and Sobolev spaces in metric spaces are also proved.