A (small) regularity zoo


Hoelder C^k Schwarz Sobolev L^p Hardy Orlicz

The image is a imagemap (which doesn’t scale because doesn’t allow Javascript widgets :/ and so everything probably looks off on your screen): move your cursor around it and explore!

Some general (imprecise) notes:

  • Function spaces in the left two columns are defined by regularity conditions
  • Those in the right three columns are defined by integrability conditions (BMO, the space of functions with bounded mean oscillation, is the dual of the Hardy space H^1)
  • Sobolev spaces W^{k,p} are defined by some mix of integrability and regularity conditions.
  • Most of the spaces above are Banach spaces; some (e.g. L^2) are Hilbert spaces.
  • Functions in any of the above spaces are (a fortiori) measurable.
  • Sobolev spaces embed into C^k / Hoelder and L^p spaces; the precise statements can get tricky.
  • This map is in no way comprehensive, it only sketches some links between some common families.