Assume that the local Lie algebra (W, Ω−δ s ) is in Floc .
Classification of Simple Lie Algebras on a Lattice
Therefore V (β ) should belong to the class Floc .
Classification of Simple Lie Algebras on a Lattice
F (U ) = {u ∈ Floc (U ) : ZU |u|2dµ + ZU dΓ(u, u) < ∞}, Fc (U ) = {u ∈ F (U ) : the essential support of u is compact in U }. F 0 (U ) = the closure of Fc (U ) for the norm (cid:18)ZU |u|2dµ + ZU dΓ(u, u)(cid:19)1/2 Deﬁnition 2.1.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
E s (x, y ) := sup (cid:8)f (x) − f (y ) : f ∈ Floc (X ) ∩ C (X ), dΓ(f , f ) ≤ dµ(cid:9), for al l x, y ∈ X , where C (X ) is the space of continuous functions on X .
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
Then any function f on Ω which is Lipschitz with respect to dΩ with Lipschitz constant CL is in Floc (Ω) and satisﬁes Ω pΥ(f , f ).
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
Recall that F (V ) = {u ∈ Floc (V ) : ZV |u|2dµ + ZV dΓ(u, u) < ∞}.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
Let V be open and f ∈ Fc (V )′ , the dual space of Fc (V ) (identify L2(X, µ) with its dual space using the scalar product). A function u : V → R is a local weak solution of the Laplace equation −Lu = f in V , if (i) u ∈ Floc (V ), (ii) For any function φ ∈ Fc (V ), E (u, φ) = R f φ dµ.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
Let Fc (I × V ) = {u ∈ Floc (I × V ) : u(t, ·) has compact support in V for a.e. t ∈ I }.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
Lu in Q, if (i) u ∈ Floc (Q), (ii) For any open interval J relatively compact in I , u φ dµ dt + ZJ E (u(t, ·), φ(t, ·))dt = 0. ∀φ ∈ Fc (Q), ZJ ZV Remark 4.10.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
Assume that h ∈ Floc is positive and continuous on W and satisﬁes ∀ u ∈ Fc (W ), E (f , u) = γ Z hu dµ for some γ ∈ R.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
H −1 (cid:0)Fc (W ) ∩ L∞(W, µ)(cid:1) = Fc (W ) ∩ L∞ (W, µ) follows from the fact that Floc (W ) ∩ L∞loc (W, µ) is an algebra.
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
This implies that some subsequence of (hn ) converges in U to a function h ∈ Floc (U ) which is positive and a local weak solution of −Lh = λU h in U .
The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms
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