L ¹→L ^q Poincaré Inequalities for 0 < q < 1 Imply Representation Formulas

Abstract

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B ₀⊂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 Poincaré inequality of the following form:

$$ {\int_B {{\left| {f - f_{B} } \right|}dv \leqslant cr{\left( B \right)}} }{\int_B {gd\mu } }, $$

for all metric balls B⊂B ₀⊂S, implies a variant of representation formula of fractional integral type: for ν-a.e. x∈B ₀,

$$ {\left| {f{\left( x \right)} - f_{{B_{0} }} } \right|} \leqslant C{\int_{B_{0} } {g{\left( y \right)}} }\frac{{\rho {\left( {x,y} \right)}}} {{\mu {\left( {B{\left( {x,\rho {\left( {x,y} \right)}} \right)}} \right)}}}d\mu {\left( y \right)} + C\frac{{r{\left( {B_{0} } \right)}}} {{\mu {\left( {B_{0} } \right)}}}{\int_{B_{0} } {g{\left( y \right)}d\mu {\left( y \right)}} }. $$

One of the main results of this paper shows that an L ¹ to L ^q Poincaré inequality for some 0 < q < 1, i.e.,

$$ {\left( {{\int_B {{\left| {f - f_{B} } \right|}^{q} dv} }} \right)}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \leqslant cr{\left( B \right)}{\int_B {gd\mu } }, $$

for all metric balls B⊂B ₀, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition,

$$ {\mathop {\sup }\limits_{\lambda > 0} }\frac{{\lambda \nu {\left( {{\left\{ {x \in B:{\left| {f{\left( x \right)} - f_{B} } \right|} > \lambda } \right\}}} \right)}}} {{\nu {\left( B \right)}}} \leqslant Cr{\left( B \right)}{\int_B {gd\mu } }, $$

also implies the same formula.

Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved.