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The Hausdorff dimension of elliptic and elliptic-caloric measure in $ {\bf R}\sp N,\;N\geq 3$

Author: Caroline Sweezy
Journal: Proc. Amer. Math. Soc. 121 (1994), 787-793
MSC: Primary 35J25; Secondary 30C85, 31A15, 35K20
MathSciNet review: 1186138
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Abstract: The existence of an L-caloric measure with parabolic Hausdorff dimension $ 4 - \varepsilon $ in $ {{\mathbf{R}}^2} \times {{\mathbf{R}}^1}$ is demonstrated. The method is to use a specially constructed quasi-disk Q whose boundary has Hausdorff $ \dim = 2 - \varepsilon $. There is an elliptic measure supported on the entire boundary of Q. Then the L-caloric measure on $ {\partial _p}Q \times [0,T]$ is compared with the corresponding elliptic measure. The same method gives the existence of an elliptic measure in $ {{\mathbf{R}}^n}$ whose support has Hausdorff $ \dim n - \varepsilon $ for $ n \geq 3$, and an L-caloric measure in $ {{\mathbf{R}}^n} \times {{\mathbf{R}}^1}$ supported on a set of parabolic Hausdorff dimension $ n + 2 - \varepsilon $.

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Keywords: L-caloric measure, parabolic Hausdorff dimension, NTA domains
Article copyright: © Copyright 1994 American Mathematical Society

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