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Assuming merely $j\\notin L^1(\\mathbb{R}^N)$, we show local compactness of the embedding $\\mathcal{D}^j(\\mathbb{R}^N)\\hookrightarrow L^2(\\mathbb{R}^N)$, where $\\mathcal{D}^j(\\mathbb{R}^N)$ denotes the space of functions $u\\in L^2(\\mathbb{R}^N)$ with $\\mathcal{E}_j(u,u)<\\infty$. 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Assuming merely $j\\notin L^1(\\mathbb{R}^N)$, we show local compactness of the embedding $\\mathcal{D}^j(\\mathbb{R}^N)\\hookrightarrow L^2(\\mathbb{R}^N)$, where $\\mathcal{D}^j(\\mathbb{R}^N)$ denotes the space of functions $u\\in L^2(\\mathbb{R}^N)$ with $\\mathcal{E}_j(u,u)<\\infty$. 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