Grant: Discovery Grants Program - Individual

What is proposed here is  for us to develop an analysis of the potentials & capacities for graphical hypersurfaces of smooth functions on the punctured Euclidean space based on the recent works: # D. Adams, Capacity and blow-up for the 3+1 dimensional wave operator. Forum Math. 20(2008)341-357 | Capacity and potentials on curves. Forum Math. 22(2010)985-991 | Wolff potentials and the 3-d wave operator. Forum Math. 23(2011)961-967. # J. Xiao, Entropy flux-electrostatic capacity-graphical mass. Proc. Amer. Math. Soc. 145(2017)825-832 | The transport equation in scaling invariant Besov or Essén-Janson-Peng-Xiao space. J. Differential Equations 266(2019)7124-7151 | Energy formulae for fractional Schrödinger-Poisson system. J. Differential Equations 331(2022)50-69. More precisely, given a smooth function G defined on Rn \{0} & g a smooth cut-off function g on Rn & 0 in ?Dx(0,T), give a better trade-off between noise removal & edge preservation in image analysis ? How much is the fractional calculus involved in this image processing ? Needless to say, answering the objective-based questions will not only reveal the nice behavior of a hypersurface average within wave propagation but also add new ideas-methods with the heat-wave nature to Applied Mathematics & Mathematical Physics & Information Technology.