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Settings in which many Sobolev spaces are not algebras
SPEAKER  |  Luke Rogers
DATE  |  May 12(Fri), 2017
TIME  |  14:50
PLACE  |  1503
Keyword  |  
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ABSTRACT  |  Let (X,レ)(X,レ) be a measure space on which there is a non-negative definite self-adjoint Markovian operator ?L with dense domain in L2(レ)L2(レ). The cannonical example is the (non-negative) Laplacian on Euclidean space, and in other contexts it is known that natural examples of such operators ?L can be used in place of the Laplacian to define differential equations for physical phenomena such as wave and heat propagation on the space XX. The corresponding Sobolev spaces are Wメ,p?={f#Lp:?メ/2#Lp}WLメ,p={f#Lp:Lメ/2#Lp}. In the Euclidean setting with ?=?ツL=?ツ it is well-known that the space Wメ,p?ツ○L…W?ツメ,p○L… is an algebra when p#(1,…)p#(1,…) and メ>0メ>0. There are extensions of this result to Lie groups with polynomial volume growth and manifolds with positive injectivity radius and non-negative Ricci curvature. In joint work with Thierry Coulhon we give bounds on the range of pp and メメ for which such results can be true in a general space (X,レ)(X,レ) by exhibiting a class of fractal spaces and operators for which the algebra property fails.
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