Snippets

# A (small) regularity zoo

The image is a imagemap (which doesn’t scale because WordPress.com 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.
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