On the jump behavior of distributions and logarithmic averages (Q944354)

The authors follow Łojasiewicz's definition of the value of a distribution at a point to define a useful notion in the theory of distributions, the jump behaviour of a distribution \(f\in {\mathcal D}'\) at a point \(x_0\in \mathbb R\) in the following way: \(f(x_0+\beta x)=a_-H(-x)+a_+H(x)\), \(\beta\to 0^+\) in \({\mathcal D}'\), where \(H\) is the Heaviside function and \(a_\pm\) are constants. They give formulas for the jump of distributions in terms of logarithmic averages. The case of symmetric jumps and applications of the distributional jump behavior at a point are specifically elaborated.