Softcover ISBN:  9783985470563 
Product Code:  EMSMEM/5 
List Price:  $75.00 
AMS Member Price:  $60.00 
Softcover ISBN:  9783985470563 
Product Code:  EMSMEM/5 
List Price:  $75.00 
AMS Member Price:  $60.00 

Book DetailsMemoirs of the European Mathematical SocietyVolume: 5; 2023; 138 ppMSC: Primary 46; Secondary 31; 28; 30
In the ordinary theory of Sobolev spaces on domains of \(\mathbb{R}^{n}\), the \(p\)energy is defined as the integral of \(\vert\nabla f\vert^{p}\). In this book, the author tries to construct a \(p\)energy on compact metric spaces as a scaling limit of discrete \(p\)energies on a series of graphs approximating the original space. In conclusion, the author proposes a notion called conductive homogeneity under which one can construct a reasonable \(p\)energy if \(p\) is greater than the Ahlfors regular conformal dimension of the space. In particular, if \(p = 2\), then he constructs a local regular Dirichlet form and shows that the heat kernel associated with the Dirichlet form satisfies upper and lower subGaussian type heat kernel estimates. As examples of conductively homogeneous spaces, the author presents new classes of squarebased, selfsimilar sets and rationally ramified Sierpiński crosses, where no diffusions were constructed before.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
ReadershipResearchers interested in the study of selfsimilar sets, analysis on metric spaces, potential theory on fractals and metric spaces, and in the applications of quasiconformal mappings.

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In the ordinary theory of Sobolev spaces on domains of \(\mathbb{R}^{n}\), the \(p\)energy is defined as the integral of \(\vert\nabla f\vert^{p}\). In this book, the author tries to construct a \(p\)energy on compact metric spaces as a scaling limit of discrete \(p\)energies on a series of graphs approximating the original space. In conclusion, the author proposes a notion called conductive homogeneity under which one can construct a reasonable \(p\)energy if \(p\) is greater than the Ahlfors regular conformal dimension of the space. In particular, if \(p = 2\), then he constructs a local regular Dirichlet form and shows that the heat kernel associated with the Dirichlet form satisfies upper and lower subGaussian type heat kernel estimates. As examples of conductively homogeneous spaces, the author presents new classes of squarebased, selfsimilar sets and rationally ramified Sierpiński crosses, where no diffusions were constructed before.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Researchers interested in the study of selfsimilar sets, analysis on metric spaces, potential theory on fractals and metric spaces, and in the applications of quasiconformal mappings.