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Azagra Rueda, Daniel and Ferrera Cuesta, Juan (2015) Regularization by supinf convolutions on Riemannian manifolds: An extension of LasryLions theorem to manifolds of bounded curvature. Journal of mathematical analysis and applications, 423 (2). pp. 9941024. ISSN 0022247X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X14009457
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Abstract
We show how LasryLions's result on regularization of functions defined on Rn or on Hilbert spaces by sup inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds M of bounded sectional curvature. More specifically, among other things we show that if the sectional curvature K of M satisfies K0 <= K <= K0 on M for some K0 >= 0, and if the injectivity and convexity radii of M are strictly positive, then every bounded, uniformly continuous function f : M > R can be uniformly approximated by globally C1,C1 functions defined by
(f(lambda))(mu) = sup(z is an element of M nu is an element of M)inf {f(y) + 1/2 lambda d(z, y)(2)  1/2 mu d(x, z)(2)}
as lambda, mu > 0(+), with 0 < mu < lambda/2. Our definition of (global) C1,C1 smoothness is intrinsic and natural, and it reduces to the usual one in flat spaces, but we warn the reader that, in the noncompact case, this definition differs from other notions of (rather local) C1,C1 smoothness that have been recently used by other authors (based on charts). The importance of this regularization method lies (rather than on the degree of smoothness obtained) on the fact that the correspondence f bar right arrow (f(lambda))(mu) is explicit and preserves many significant geometrical properties that the given functions f may have, such as invariance by a set of isometries, infima, sets of minimizers, ordering, local or global Lipschitzness, and (only when one additionally assumes that K <= 0) local or global convexity. We also give two examples showing that this result completely fails, even for (nonflat) CartanHadamard manifolds, whenever f or K are not bounded.
Item Type:  Article 

Uncontrolled Keywords:  approximation, convex function, semiconcave function, Lipschitz, Riemannian manifold, distance function, LasryLions regularization, C 1,1 function. 
Subjects:  Sciences > Mathematics > Differential geometry 
ID Code:  33309 
Deposited On:  29 Sep 2015 11:21 
Last Modified:  29 Sep 2015 11:21 
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