Summary
This book is a study of linear, self-adjoint, second order, elliptic differential operators. The goal is to investigate spectral properties and obtain pointwise bounds on eigenfunctions by studying the heat kernels.
There is an enormous literature on heat kernels which stretches back half a century, so it is easy to imagine that the subject has already reached its final form. However, we shall make almost no reference to this literature, and shall rely entirely upon results proved within the last five years using quadratic form techniques and logarithmic Sobolev inequalities. These new techniques have led to radically better global bounds on the heat kernels.
We shall be concerned to obtain pointwise upper and lower bounds on various functions in terms of effectively computable constants. In a number of cases the new methods yield constants which are sharp or at least of the correct order of magnitude. This is in sharp distinction to much of the older theory, where various constants appeared to depend upon the magnitudes of the derivatives of the second order coefficients of the differential operator, although in fact they do not.
Because of our approach we are able to deal simply and naturally with operators in divergence form whose second order coefficients are measurable. Earlier treatments of this problem such as that of Gilbarg and Trudinger have relied heavily upon Moser's Harnack inequality. In spite of its fundamental historical and conceptual importance, we make no mention of Moser's approach.
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- Heat Kernels and Spectral Theory , pp. vii - xPublisher: Cambridge University PressPrint publication year: 1989