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Conde-Alonso, Mihalis Mourgoglou, Xavier Tolsa","submitted_at":"2018-01-25T15:35:23Z","abstract_excerpt":"Consider a totally irregular measure $\\mu$ in $\\mathbb{R}^{n+1}$, that is, the upper density $\\limsup_{r\\to0}\\frac{\\mu(B(x,r))}{(2r)^n}$ is positive $\\mu$-a.e.\\ in $\\mathbb{R}^{n+1}$, and the lower density $\\liminf_{r\\to0}\\frac{\\mu(B(x,r))}{(2r)^n}$ vanishes $\\mu$-a.e. in $\\mathbb{R}^{n+1}$. We show that if $T_\\mu f(x)=\\int K(x,y)\\,d\\mu(y)$ is an operator whose kernel $K(\\cdot,\\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\\\"older continuous coefficients, then $T_\\mu$ is not bounded in $L^2(\\mu)$. 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Conde-Alonso, Mihalis Mourgoglou, Xavier Tolsa","submitted_at":"2018-01-25T15:35:23Z","abstract_excerpt":"Consider a totally irregular measure $\\mu$ in $\\mathbb{R}^{n+1}$, that is, the upper density $\\limsup_{r\\to0}\\frac{\\mu(B(x,r))}{(2r)^n}$ is positive $\\mu$-a.e.\\ in $\\mathbb{R}^{n+1}$, and the lower density $\\liminf_{r\\to0}\\frac{\\mu(B(x,r))}{(2r)^n}$ vanishes $\\mu$-a.e. in $\\mathbb{R}^{n+1}$. We show that if $T_\\mu f(x)=\\int K(x,y)\\,d\\mu(y)$ is an operator whose kernel $K(\\cdot,\\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\\\"older continuous coefficients, then $T_\\mu$ is not bounded in $L^2(\\mu)$. 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