*Jean-Michel Bismut*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691151298
- eISBN:
- 9781400840571
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151298.003.0013
- Subject:
- Mathematics, Geometry / Topology

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 ...
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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.Less

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*.

*Jean-Michel Bismut*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691151298
- eISBN:
- 9781400840571
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151298.003.0014
- Subject:
- Mathematics, Geometry / Topology

This chapter obtains uniform bounds for the kernels rb,tX and another rb,tX for bounded b > 0, with the proper decay at infinity on X or ̂X. These bounds will be used to obtain corresponding bounds ...
More

This chapter obtains uniform bounds for the kernels rb,tX and another rb,tX for bounded b > 0, with the proper decay at infinity on X or ̂X. These bounds will be used to obtain corresponding bounds for the kernel qb,tX in the next chapter. Furthermore, the arguments developed in Chapter 12, which connect the hypoelliptic heat kernel with the wave equation, play an important role in proving the required estimates. Hence, this chapter first establishes estimates on the Hessian of the distance function on X. Then, the chapter obtains bounds on the first heat kernel rb,tX and establishes the bounds on another heat kernel rb,tX.Less

This chapter obtains uniform bounds for the kernels rb,tX and another rb,tX for bounded *b* > 0, with the proper decay at infinity on *X* or ̂X. These bounds will be used to obtain corresponding bounds for the kernel qb,tX in the next chapter. Furthermore, the arguments developed in Chapter 12, which connect the hypoelliptic heat kernel with the wave equation, play an important role in proving the required estimates. Hence, this chapter first establishes estimates on the Hessian of the distance function on *X*. Then, the chapter obtains bounds on the first heat kernel rb,tX and establishes the bounds on another heat kernel rb,tX.

*Isroil A. Ikromov and Detlef Müller*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691170541
- eISBN:
- 9781400881246
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691170541.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's ...
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This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's interpolation theorem for analytic families of operators. It constructs analytic families of complex measure μsubscript Greek small letter zeta, for ζ in the complex strip Σ given by 0 ≤ Reζ ≤ 1, by introducing complex coefficients in the sums defining the measures νsubscript Greek small letter delta,jsuperscript V and νsubscript Greek small letter delta,jsuperscript V I, respectively. These coefficients are chosen as exponentials of suitable affine-linear expression in ζ in such a way that, in particular, μsubscript Greek small letter theta subscript c = νsubscript Greek small letter delta,jsuperscript V I, respectively, μsubscript Greek small letter theta subscript c = νsubscript Greek small letter delta,jsuperscript V I. As it turns out, the main problem consists in establishing suitable uniform bounds for the measure μsubscript Greek small letter zeta when ζ lies on the right boundary line of Σ.Less

This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's interpolation theorem for analytic families of operators. It constructs analytic families of complex measure μsubscript Greek small letter zeta, for ζ in the complex strip Σ given by 0 ≤ Reζ ≤ 1, by introducing complex coefficients in the sums defining the measures νsubscript Greek small letter delta,*j*superscript *V* and νsubscript Greek small letter delta,*j*superscript *V I*, respectively. These coefficients are chosen as exponentials of suitable affine-linear expression in ζ in such a way that, in particular, μsubscript Greek small letter theta subscript *c* = νsubscript Greek small letter delta,*j*superscript *V I*, respectively, μsubscript Greek small letter theta subscript *c* = νsubscript Greek small letter delta,*j*superscript *V I*. As it turns out, the main problem consists in establishing suitable uniform bounds for the measure μsubscript Greek small letter zeta when ζ lies on the right boundary line of Σ.