A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces
Let \((S,\rho,\mu)\) be a quasimetric space of homogeneous type endowed with a doubling Borel measure \(\mu\) and assume the following geometric hypothesis: Given a fixed ball \(B_0\) in \((S,\rho)\), for each \(x\in B_0\) there exists a chain of balls \(B_j(x)\), \(j=1,2,\dots\), such that, for all \(j\geq0\), (H1) \(B_j(x)\subset B_0\); (H2) \(r(B_j(x))\approx 2^{-j}r(B_0)\); (H3) \(\rho(B_j(x),x)\leq cr(B_j(x))\). Let \(a:B\to a(B)\) be a nonnegative functional defined on balls \(B\subset B_0\). Consider the sum operator \(T\) associated with the functional \(a\) defined by \(T(x)=\sum_{j=0}^\infty a(B_j(x))\), \(x\in B_0\), where \(\{B_j(x)\}_{j=1}^\infty\) is a chain of balls satisfying (H1)--(H3) and \(B_0(x)=B_0\) for every \(x\in B_0\). The main purpose of the present paper is to show that the sum operator \(T\) plays the role of integral operators of Riesz-potential type in proving Poincaré-Sobolev estimates. This can be illustrated by one of the main results which establishes the pointwise representation formula: If \(f\in L^1(B_0,\mu)\) is such that for any ball \(B\subset B_0\), \(| B_\mu| ^{-1}\int_B| f-f_B| d\mu\leq ca(B)\), then for \(\mu\)-a.a. \(x\in B_0\), \(| f(x)-f_{B_0}| \leq CT(x)\). Further results include the weak estimate for \(T\), the weak self-improving property of the Poincaré inequality in \(B_0\), the two-weight estimate and the case \(p=q\). The paper improves some earlier results, namely by the authors [J. Funct. Anal. 153, No. 1, 108--146 (1998; Zbl 0892.43005)] and by \textit{P. Hajłasz} and \textit{P. Koskela} [Mem. Am. Math. Soc. 688 (2000; Zbl 0954.46022)].
scientific article; zbMATH DE number 2102104