Fractional Sobolev Embedding Theorem

Theorem

Let $S'$ denote the space of tempered distributions.

Let $\FF : S' \to S'$ denote the Fourier transform.

For each $s \in \R$ and $p \in \closedint 1 \infty$, let:

$\map {W^{s, p} } {\R^n} = \set {u \in S': \sequence \xi^s \hat u \in \map {L^p} {\R^n} }$

where:

$\sequence \xi = \paren {1 + \size \xi^2}^\frac 1 2$

Then:

$(1): \quad$ If $s > t$ then $\map {W^{s, p} } {\R^n}$ embeds continuously into $\map {W^{t, q} } {\R^n}$ where $q$ is given by $\dfrac 1 q = \dfrac 1 p - \dfrac {s - t} n$.

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Source of Name

This entry was named for Sergei Lvovich Sobolev.