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Suppose<i>X</i>and<i>Y</i>are locally compact Hausdorff spaces,<i>E</i>and<i>F</i>are Banach spaces, and<i>F</i>is strictly convex. We show that every linear isometry<i>T</i>from<i>C</i>₀(<i>X</i>,<i>E</i>)<i>into</i><i>C</i>₀(<i>Y</i>,<i>F</i>) is essentially a weighted composition operator<i>Tf</i>(<i>y</i>)=<i>h</i>(<i>y</i>)(<i>f</i>((<i>y</i>))).

In this paper, we generalize the Cao-Yau'’s gradient estimate for the sum of squares of vector …elds up to higher step under ssumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel. 1.

No abstract uploaded!

No abstract uploaded!