scholarly journals Exponential coordinates and regularity of groupoid heat kernels

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Bing So
Keyword(s):  
Differential Operator ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Pseudo Differential Operator

AbstractWe prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.

The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2457-2469
Author(s):  
Akhilesh Prasad ◽  
S.K. Verma
Keyword(s):  
Differential Operator ◽  
Function Spaces ◽  
Pseudo Differential Operator ◽  
Test Function ◽  
Basic Properties ◽  
Cone Function ◽  
Translation Operators

In this article, weintroduce a new index transform associated with the cone function Pi ??-1/2 (2?x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp(I, x-1/2 dx) norm. The test function spaces G? and F? are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between G? and F?.

scholarly journals Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator. The method of positive commutators

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Abdelwaheb Ifa ◽  
Michel Rouleux
Keyword(s):  
Differential Operator ◽  
Gram Matrix ◽  
Pseudo Differential Operator ◽  
Quantization Rule ◽  
International Audience ◽  
Scalar Products ◽  
Quantization Rules

International audience We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the " flux norm " is not invertible. Dans le cadre algébrique et microlocal élaboré par Helffer et Sjöstrand, on propose une ré-écriture de la règle de quantification de Bohr-Sommerfeld pour un opérateur auto-adjoint h-Pseudo-différentiel 1-D; elle s'exprime par la non-inversibilité de la matrice de Gram d'un couple de solutions WKB dans une base convenable, pour le produit scalaire associé à la " norme de flux " .

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

A proof of the main identity

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Heat Kernel ◽  
Heat Kernels

This chapter proves the formula that was stated in Chapter 6. It first states various estimates on the hypoelliptic heat kernels, which are valid for b ≥ 1 and then makes a natural rescaling on the coordinates parametrizing ̂X. Next, the chapter introduces a conjugation on the Clifford variables and shows that the norm of the term defining the conjugation can be adequately controlled. The chapter then introduces a conjugate ℒA,bX of another ℒA,bX and its associated heat kernel. Afterward, the chapter obtains the limit as b → +∞ of the rescaled heat kernel, thus establishing the formula in Chapter 6. After further computations, this chapter states a result on convergence of heat kernels.

Heat Kernels of Lorentz Cones

1999 ◽  
Vol 42 (2) ◽  
pp. 169-173 ◽  
Author(s):  
Hongming Ding
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Lower Bounds ◽  
Heat Kernels ◽  
Upper And Lower Bounds ◽  
Symmetric Cones ◽  
Lorentz Cone

AbstractWe obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time t and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

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