# Compact embedding of fractional space

Is the space $$H^\lambda((0,T); H^1(K))$$ for $$0 <\lambda <1$$ where $$K$$ is compact subset of $$\mathbb{R}^n$$ compactly embedded in $$L^2( (0,T) \times K)$$?

$$H^\lambda((0,T); H^1(K))\hookrightarrow \hookrightarrow L^2((0,T) ; H^1(K))\hookrightarrow \hookrightarrow L^2((0,T); L^2(K))$$. Where can I find these type of results.

There are (at least) two ways to prove $$H^\gamma(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;L^2).$$

1. $$H^\gamma(0,T;H^1) \hookrightarrow H^\gamma(0,T;(H^1)') \cap L^2(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;L^2).$$
2. $$H^\gamma(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;H^1) \hookrightarrow L^2(0,T;L^2).$$

Note that you don't get that $$L^2(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;L^2)$$ as stated in the question. This is not true. For example take the sequence $$f_n(t,x)=\sin(nt)x$$.

First: Ben Schweizer "Partielle Differentialgleichungen", Theorem 24.2

Let $$V,X,V_0$$ be Hilbert spaces such that $$V \hookrightarrow \hookrightarrow X \hookrightarrow V_0$$. Then for $$T<\infty$$ and $$\gamma>0$$ we have $$H^\gamma(0,T;V_0) \cap L^2(0,T;V) \hookrightarrow \hookrightarrow L^2(0,T;X).$$

Second: You get $$H^\gamma(0,T) \hookrightarrow \hookrightarrow L^2(0,T)$$ from the fractional Sobolev embedding.

• Is it also true that $H^\lambda((0,T);H^1(\mathbb{R}^d))$ is continuously embedded in $C((0,T); L^2(\mathbb{R}^d))$ – MathAnimal May 6 at 15:19
• @MathAnimal It's true at least for $\lambda >1/2$, see also this question. – Fritz May 6 at 15:30
• Not for $\lambda <1/2$? – MathAnimal May 6 at 15:38