ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract LetU be a basis representation of an irreducible unitary representation of a nilpotent Lie groupG inL 2(R k) and letdU denote the representation of the Lie algebrag obtained by differentiation. Ifb 1,...,b d is a basis ofg andB i =dU(b i ) we consider the operators $$H = - \sum\limits_{i,j = 1}^d {c_{ij} B_i B_j + } \sum\limits_{i = 1}^d {c_i B_i } ,$$ whereC=(c ij ) is a real symmetric strictly positive matrix andc i ∈C. ThenH generates a continuous semigroupS, holomorphic in the open right half-plane, with a reduced kernek κ defined by $$(S_z \varphi )(x) = \int\limits_{R^k } {dy\kappa _z (x;y)\varphi (y).} $$ We prove Gaussian off-diagonal bounds and “exponential” on-diagonal bounds for κ. For example, ifc i =0 we establish that $$\left| {\kappa _t (x;y)} \right| \leqq a(1 \wedge \varepsilon \mu t)^{ - {k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} e^{ - \lambda _1 t} e^{ - d(x;y)^2 (4(1 + \varepsilon )t)^{ - 1} } $$ for allt〉0 and ɛ ∈ 〈0,1], where μ is the smallest eigenvalue ofC, λ1 is the smallest eigenvalue ofH andd is a natural distance associated with the coefficientsC and the representationU. Bounds are also obtained forc i ≠) and complext. Alternatively, ifH is self-adjoint then $$\left| {\kappa _z (x;y)} \right| \leqq ae^{ - \lambda _1 \operatorname{Re} z} e^{ - b(\left| x \right|^\alpha + \left| y \right|^\alpha )} $$ for allz ∈C with Rez ≧ 1, for some α ∈ 〈0,2].
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02101655
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