Sobolev embedding $W^{1,2}(\Omega)\subset L^p(\Omega)$ where $\Omega$ is a halfplane

Question

I would like to ask when the following Sobolev embedding holds true

$$W^{1,2}(\Omega)\subset L^p(\Omega)$$

where $\Omega\subset \mathbb{R}^2$ is any open set and $1 < p < \infty$. All book references concerns the case when the dimension of domain is strictly less or greater than $2$.

Actually, I ask this question because I need the following embedding

$$W^{1,2}(\mathbb{R}^2_+)\subset L^4(\mathbb{R}^2_+)$$ where $\mathbb{R}^2_+$ is the upper half plane.

Answer 1

This is true, yes, you actually have $$ W^{1,2}(\mathbb R_+^2)\subset L^q(\mathbb R_+^2) $$ for any $2\leq q<\infty$.

This is the case of embedding $p=n$.