Muller.-.Functional.Calculus.on.Lie.Groups.(1998).[sharethefiles.com].pdf

elds which generate the Lie algebra g of G Moreover for simplicity we shall

X of X g is given by X so that by a straightforward extension

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L ux t on G R

whose solution is given by u tcost

on R

Fix a nontrivial cuto function C

R supported in the interval

k m k

jj mr jj

R denotes the Sobolevspace of order Thus kmk

so that mLf f K

for every n N where jAj denotes the Haar measure of a

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j U

j Ce

for every n

algebra g admits a decomposition into subspaces

g g

such that g

for all i k and where g

generates g as a Lie algebra We

of g

j dim g

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k m k

for some Q then mL is boundedonL

G for p

If m L

it is easy to see that mL is a convolution operator mLf f K

dy where a priori the convolution kernel K

Observe that in and in comparizon to the classical case G R

Fix n N and let H

d n for which the group law expressed in coordinates x y u R

where x y denotes the Euclidean inner product A basis of the Lie algebra of H

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Theorem For the subLaplacian L on H

norm on H

gj rjgj for every g H

j j g j

g j

dg C jj m jj

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