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TY - ABST
AU - Ceccon, Jurandir
AU - Montenegro, Marcos
TI - Homogeneous sharp Sobolev inequalities on product manifolds
JO - Proceedings Section A: Mathematics - Royal Society of Edinburgh
PY - 2006-05-01T00:00:00///
VL - 136
IS - 2
SP - 277
EP - 300
N2 - Let (*M, g*) and (*N, h*) be compact Riemannian manifolds of dimensions *m* and *n*, respectively. For *p*-homogeneous convex functions *f*(*s, t*) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on *H*^{1, p}(*M* × *N*)

‖*u*‖^{p}_{Lp* (M × N)} ≤ *K*^{p}_{f}‖*f*(|∇*u*|_{g},|∇*u*|_{h})‖_{L}^{1}_{(M × N)} + *B*‖*u*‖^{p}_{Lp(M × N)},

where

1 < *p* < *m* + *n*, *p** = (*m* + *n*)*p*/*m* + *n* − *p*

and *K*_{f} = *K*_{f} (*m, n, p*) is the best constant of the homogeneous Sobolev inequality on *D*^{1, p} (R^{m+n}),

‖*u*‖^{p}_{Lp*(Rm+n)} ≤ *K*^{p}_{f}‖*f*(|∇_{x}*u*|,|∇_{y}*u*|)‖_{L1(Rm+n)}.

The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions *f*, we investigate the validity in a uniform sense on *f* and argue with suitable approximations of *f* which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.
UR - http://www.ingentaconnect.com/content/rse/proca/2006/00000136/00000002/art00003
ER -