IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

On the Harmonic Analysis of the Space BV.

  • Nov. 6, 2000, 2:15 p.m. LeConte 312
  • Nov. 13, 2000, 2:15 p.m. LeConte 312
  • Nov. 20, 2000, 2:15 p.m. LeConte 312
  • Nov. 27, 2000, 2:15 p.m. LeConte 312

Abstract

The space BV of functions of bounded variation in several variables plays a pivotal role in many application areas. It is frequently used as a model for real life images; the solution operator for hyperbolic conservation laws is a contraction on BV which is at the center of the regularity theory for these nonlinear PDEs; in differential geometry BV appears in the isoperemetric inequalities; in regularity for PDEs it occurs in the Gagliardo-Nirenberg inequalities. The structure of BV is complicated by the fact that it does not admit an unconditional basis.

These lectures will unfold new structure theorems for BV by using wavelet decompositions.

We prove a sandwich theorem for the wavelet coefficients of BV functions which can be used in many cases as a substitute for an unconditional basis. Using these sandwich theorems we show how to prove new fundamental results concerning BV in the interpolation of linear operators. We then apply these interpolation theorems to the application areas mentioned above.

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