We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is "almost" characterized by wavelet expansions in the following sense: if a function is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak- type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev or Besov spaces, and to derive new Gagliardo-Nirenberg-type inequalities.
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Albert Cohen, Wolfgang Dahmen, Ingrid Daubechies, Ronald A. DeVore, Harmonic Analysis of the space BV. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 235–263DOI 10.4171/RMI/345