Abstract
The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L 2 norm to the infimal convolution of the L∞ and H -1 norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 129-144 |
| Number of pages | 16 |
| Journal | Bulletin of the London Mathematical Society |
| Volume | 51 |
| Issue number | 1 |
| DOIs | |
| State | Published - Feb 1 2019 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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A dual form of the sharp Nash inequality and its weighted generalization. / Carlen, Eric; Lieb, Elliott H.
In: Bulletin of the London Mathematical Society, Vol. 51, No. 1, 01.02.2019, p. 129-144.Research output: Contribution to journal › Article
TY - JOUR
T1 - A dual form of the sharp Nash inequality and its weighted generalization
AU - Carlen, Eric
AU - Lieb, Elliott H.
PY - 2019/2/1
Y1 - 2019/2/1
N2 - The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L 2 norm to the infimal convolution of the L∞ and H -1 norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.
AB - The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L 2 norm to the infimal convolution of the L∞ and H -1 norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.
UR - http://www.scopus.com/inward/record.url?scp=85056705442&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85056705442&partnerID=8YFLogxK
U2 - 10.1112/blms.12220
DO - 10.1112/blms.12220
M3 - Article
VL - 51
SP - 129
EP - 144
JO - Bulletin of the London Mathematical Society
JF - Bulletin of the London Mathematical Society
SN - 0024-6093
IS - 1
ER -