Potentials

We recall that, if y ∈ ℝN, then the function defined by % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGvbWdamaaBaaaleaapeGaamyEaaWdaeqaaOWdbmaabmaapaqa % a8qacaWG4baacaGLOaGaayzkaaGaeyypa0JaeyOeI0IaciiBaiaac+ % gacaGGNbWaauWaa8aabaWdbiaadIhacqGHsislcaWG5baacaGLjWUa % ayPcSdWdaiaaywW7peWaaeWaa8aabaWdbiaadIhacqGHGjsUcaWG5b % Gaai4oaiaad6eacqGH9aqpcaaIYaaacaGLOaGaayzkaaWaauWaa8aa % baWdbiaadIhacqGHsislcaWG5baacaGLjWUaayPcSdWdamaaCaaale % qabaWdbiaaikdacqGHsislcaWGobaaaOWdaiaaywW7caaMf8+dbmaa % bmaapaqaa8qacaWG4bGaeyiyIKRaamyEaiaacUdacaWGobGaeyyzIm % RaaG4maaGaayjkaiaawMcaaiabgUcaRiabg6HiL+aacaaMf8UaaGzb % VlaaywW7caaMf8UaaGjbV-qadaqadaWdaeaapeGaamiEaiabg2da9i % aadMhaaiaawIcacaGLPaaaaaa!74FB!$$ {U_y}\left( x \right) = - \log \left\| {x - y} \right\|\quad \left( {x \ne y;N = 2} \right){\left\| {x - y} \right\|^{2 - N}}\quad \quad \left( {x \ne y;N \geqslant 3} \right) + \infty \quad \quad \quad \quad \;\left( {x = y} \right)$$ is superharmonic on ∝ N and harmonic on ∝ N\{y}