Suppose $f_{k} \to f$ strongly in $L^{2}(\mathbb{R}^{n})$. Let $2 \le 2q < \eta$ where $\eta = \infty$ if $n \le 2$ and $\eta = 2n/(n-2)$ if $n \ge 3$. During a proof, my professor wrote that $\|f_{k}-f\|_{L^{2q}(\mathbb{R}^{n})} \to 0$ as $k \to \infty$ by interpolation inequality.
I searched the term interpolation inequality on the internet and found a lot of different results. What does it mean in the present context?
Edit: Let me add more details about my problem. I am trying to prove the following result: Let $g \in L^{p}(\mathbb{R}^{n})$ with $p > \max\{n/2,1\}$ and suppose $f_{n} \rightharpoonup f$ on $H^{1}(\mathbb{R}^{n})$, meaning that $\langle f_{n}, h\rangle_{L^{2}}+\langle \nabla f_{n},\nabla h\rangle_{L^{2}} \to \langle f, h\rangle_{L^{2}}+\langle \nabla f, \nabla h\rangle_{L^{2}}$ for every $h \in H^{1}(\mathbb{R}^{n})$. Then: $$\lim \int_{\mathbb{R}^{n}}g(x)|f_{n}(x)|^{2}dx = \int_{\mathbb{R}^{n}} g(x)|f(x)|^{2}dx$$
The proof goes like this. Let $q$ be such that $1/p+1/q = 1$. Then, by Hölder and Minkowski inequalities: $$\int_{\mathbb{R}^{n}}|g(x)|||f_{n}(x)|^{2}-|f(x)|^{2}|dx \le \int_{\mathbb{R}^{n}}|g(x)||f_{n}(x)-|f(x)|(|f_{n}(x)|+|f(x)|)dx \le \|f_{n}-f\|_{L^{2q}}\||f_{n}|+|f|\|_{L^{2q}}\|g\|_{L^{p}} \le \|f_{n}-f\|_{L^{2q}}(\|f_{n}\|_{L^{2q}}+\|f\|_{L^{2q}})\|g\|_{L^{p}} \le K\|f_{n}-f\|_{L^{2q}}$$ and the latter must go to zero by some interpolation inequality.
By interpolation of $L^p$ spaces (see here, Lemma 9), we have
$$\|f_k-f\|_{L^{2q}}\leq \|f_k-f\|_{L^\eta}^\theta \|f_k-f\|_{L^2}^{1-\theta}$$
For some exponent $0<\theta<1$ depending on $q$ that’s unimportant here. It now suffices to bound the $L^\eta$ norm of $f_k-f$, which can be derived from the Gagliardo-Nirenberg Sobolev inequality, provided that $n\geq 3$:
$$\|f_k-f\|_{L^\eta} \leq C \|f_k-f\|_{H^1}$$
The right hand side bounded since weakly convergent sequences and their limits are bounded.
However, when $n=2$, i.e. $\eta =\infty$, the Gagliardo-Nirenberg embedding is false, so I’m not sure how that’s supposed to be handled.