### Abstract

Let A be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose that A is formally self-adjoint and has positive definite leading symbol. For example, A could be the conformal Laplacian (Yamabe operator)L, or the square of the Dirac operator[Figure not available: see fulltext.]. Within the conformal class {Mathematical expression} of an Einstein, locally symmetric "background" metric g_{o} on a compact four-manifold M, we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant of A and the volume of g imply bounds on the W^{2,2} norm of the conformal factor w, provided that a certain conformally invariant geometric constant k=k(M, g_{o}A) is strictly less than 32π^{2}. We show for the operators L and[Figure not available: see fulltext.] that indeed k < 32π^{2} except when (M, g_{o}) is the standard sphere or a hyperbolic space form. On the sphere, a centering argument allows us to obtain a bound of the same type, despite the fact that k is exactly equal to 32π^{2} in this case. Finally, we use an inequality of Beckner to show that in the conformal class of the standard four-sphere, the determinant of L or of[Figure not available: see fulltext.] is extremized exactly at the standard metric and its images under the conformal transformation group O(5,1).

Original language | English (US) |
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Pages (from-to) | 241-262 |

Number of pages | 22 |

Journal | Communications In Mathematical Physics |

Volume | 149 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 1992 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*149*(2), 241-262. https://doi.org/10.1007/BF02097624