Mathematics Colloquia and Seminars

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Let $L^{m,p}(\mathbb{R}^n)$ be the Sobolev space consisting of all functions with finite seminorm $\|F\|_{L^{m,p}(\mathbb{R}^n)} = \max_{|\alpha| = m} \left\{ \| \partial^\alpha F \|_{L^p(\mathbb{R}^n)}\right\}$. Let $\mu$ be a Borel regular measure and $f$ be a $\mu$-measurable function. Under what conditions is $\inf_{F \inL^{m,p}(\mathbb{R}^n)} \{ \|F\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |F-f|^p d\mu \}^{1/p}$ finite? Can we construct an "approximate extension" operator $T$, satisfying

$(\|Tf\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |Tf-f|^p d\mu)^{1/p} \leq C \cdot \inf_{F \in L^{m,p}(\mathbb{R}^n)} \{ \|F\|_{L^{m,p}(\mathbb{R}^n)}^p +\int_{\mathbb{R}^n} |F-f|^p d\mu \}^{1/p}$,

with $C$ dependent on $m$, $n$, and $p$ only?

We'll begin by discussing interpolation and extension theory broadly before focusing on new challenges faced in answering these questions. As a consequence of this work, we determine criteria for $f: E \to \mathbb{R}$ ($E \subset \mathbb{R}^n$ arbitrary) to be the trace of a function in $L^{m,p}(\mathbb{R}^n)$, building on the work of Fefferman, Luli, and Israel.

Zoom link: https://ucdavis.zoom.us/j/91748527825?
Passcode: first six digits of pi