A local version of Hardy spaces associated with operators on metric spaces
Let \((X,d,\mu)\) a metric space, where \(d\) is a distance and \(\mu\) satisfies the doubling property. For \(L\) a non-negative and self-adjoint operator defined on \(L^2(X)\), let us consider the semigroup \(e^{-tL}\) generated by \(L\) on \(L^2(X)\) and whose kernel \(p_t(x,y)\) satisfies a Gaussian upper bound. The local Hardy space \(h^1_L(X)\) is defined as \(\{f\in L^2(X): \;\|s_{loc,L}f\|_{L^1(X)}<\infty\}\) where \(s_{loc,L}f\) is the following area function \[ s_{loc,L}f(x)=\left(\int_0^1 \int_{d(x,y)\leq t} |t^2 L e^{-t^2L} f(y)|^2 \frac{d\mu(y)}{V(x,t)}\frac{dt}{t}\right)^{1/2} \quad x\in X \] and \(V(x,t)=\mu(B(x,t))\). An atomic decomposition in terms of local \((1,p)\) atoms of this local Hardy space is proved. Also, a Moser type local boundedness condition for \(L\) is used to show that the local Hardy space \(h^1_L(X)\) can be characterized in terms of Littlewood-Paley functions. The work concludes with some applications to the study of second-order operators defined on Lipschitz domains and a class of Schrödinger operators which satisfy both, a Gaussian upper bound for the kernel, and a Moser type local condition.
scientific article; zbMATH DE number 6151455
scientific article