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Friday, September 18, 2015

**Abstract:** The Sobolev space $W^{1,p}(\mathbb{R}^n)$ is defined as the space of $L^p$ functions on $\mathbb{R}^n$ with weak derivatives in $L^p$. An equivalent way to define this space is as the completion of the space of smooth functions in $L^p$ with $L^p$-integrable gradient, using a suitable norm. Now suppose we replace $\mathbb{R}^n$ and Lebesgue measure with a more general metric space and measure. How can one define Sobolev spaces without the notion of partial derivatives? Based on a paper by Piotr Hajlasz, my talk will describe a few ways to define Sobolev spaces on metric-measure spaces and under what conditions on the space these definitions coincide.