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2014

A Further Remark on Sobolev Spaces. The Case $0<p<1$

Wennman, Aron

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We discuss a phenomenon observed by Jaak Peetre in the seventies: for small $L^{p}$-exponents, i.e.

for $0<p<1$, the Sobolev spaces $W^{k,p}$ defined in a seemingly natural way are isomorphic to $L^{p}$.
This says that the dual of $W^{k,p}$ is trivial, and indicates that these spaces are highly pathological.
In this note we expand on Peetre's observation, explaining in detail some points that might merit further discussion.

A Further Remark on Sobolev Spaces. The Case $0<p<1$ · Around

Built on

Mahlon M. Day, The spaces L p L^{p} with 0 < p < 1 0<p<1 , Bull. Amer. Math. Soc. 46
1940
Earlier work this paper cites.
J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The theory of splines and their applications , Academic Press, New York-London, 1967. MR 0239327 (39 #684)
1967
Earlier work this paper cites.

Similar

Jaak Peetre, A remark on Sobolev spaces. The case 0 < p < 1 0<p<1 , J. Approximation Theory 13
1975
Cited alongside, same era.
Gustav Behm and Aron Wennman, Carleman-Sobolev classes for small exponents , pre-print (2014)
2014
Cited alongside, same era.

Then

H. Hedenmalm and A. Wennman, A critical topology for l p l^{p} -carleman classes with 0 < p < 1 0<p<1 . , 1708.08840 (2017)
2017
Closest in time.

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