L1→Lq Poincare inequalities for 0 < q < 1 imply representation formulas

Given two doubling measures p and v in a metric space (S, p) of homogeneous type, let B-0 subset of S be a given ball. It has been a well-known result by now (see [1-4]) that the validity of an L-1 --> L-1 Poincare inequality of the following form: integral(B)\f - f(B)\dnu less than or equal to cr(B) integral(B)gdmu, for all metric balls B subset of B-0 subset of S, implies a variant of representation formula of fractional intergral type: for nu-a.e. x is an element of B-0, \f(x) - f(B0)\ less than or equal to C integral(B0) g(y) (rho(x,y))/(mu(B(x,rho(x,y)))) dmu(y) + C-r(B0)/(mu(B0)) integral(B0) g(y) dmu(y). One of the main results of this paper shows that an L-1 to L-q Poincare inequality for some 0 < q < 1, i.e., (integral(B) \f - f(B)\(q) dnu)(1/q) less than or equal to cr(B) integral(B) gdmu, for all metric balls B subset of B-0, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, sup(lambda>0) (lambdanu({x is an element of B : \f(x) - fB\ > lambda}))/(nu(B) less than or equal to Cr(B)) integral(B) g dmu, also implies the same formula. Analogous theorems related to high-order Poincare inequalities and Sobulev spaces in metric spaces are also proved.