245C, Notes 4: Sobolev spaces

Very nice article!

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As discussed in previous notes, a function space norm can be viewed as a means to rigorously quantify various statistics of a function $latex {f: X \rightarrow {\bf C}}&fg=000000$. For instance, the “height” and “width” can be quantified via the $latex {L^p(X,\mu)}&fg=000000$ norms (and their relatives, such as the Lorentz norms $latex {\|f\|_{L^{p,q}(X,\mu)}}&fg=000000$). Indeed, if $latex {f}&fg=000000$ is a step function $latex {f = A 1_E}&fg=000000$, then the $latex {L^p}&fg=000000$ norm of $latex {f}&fg=000000$ is a combination $latex {\|f\|_{L^p(X,\mu)} = |A| \mu(E)^{1/p}}&fg=000000$ of the height (or amplitude) $latex {A}&fg=000000$ and the width $latex {\mu(E)}&fg=000000$.

However, there are more features of a function $latex {f}&fg=000000$ of interest than just its width and height. When the domain $latex {X}&fg=000000$ is a Euclidean space $latex {{\bf R}^d}&fg=000000$ (or domains related to Euclidean spaces, such as open subsets of $latex {{\bf R}^d}&fg=000000$, or manifolds), then another important feature of such functions (especially in PDE)…

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