Let \(-\Delta_ D\) be the Dirichlet Laplacian for an open set D in \({\mathbb{R}}^ m\) and let \(p_ D(x,y;t)\) be the heat kernel associated to the parabolic operator \(-\Delta_ D+\partial /\partial t\). Define \(Z_ D(t)=\int_ Ddx p_ D(x,x;t)\) and \(Q_ D(t)=\int_ Ddx \int_ Ddy p_ D(x,y;t)\). Necessary and sufficient geometrical conditions on D are obtained for \(Z_ D(t)\) or \(Q_ D(t)\) to be finite for some \(t\geq 0.\)
It is conjectured that if \(0\leq s<\infty\) then the following are equivalent. (i) \(Q_ F(t)<\infty\) for all \(t\geq s\), (ii) \(Z_ F(t)<\infty\) for all \(t>s\). The conjecture holds for horn-shaped regions in \({\mathbb{R}}^ 2\) and it is not possible to sharpen it to the case where \(t=s.\)
Finally is is proved that if D is an open, bounded and connected set in \({\mathbb{R}}^ m\), \(m=2,3,\cdot \cdot \cdot\) with a smooth boundary \(\partial D\), then for all \(t\geq 0\) \[ Q_ D(t)-| D| +2t^{1/2}/\pi^{1/2}| \partial D| \quad \leq \quad Ct, \] where \(| D|\) is the volume of D, \(| \partial D|\) is the area of the boundary \(\partial D\) and C is a constant depending upon the smoothness of the boundary.