Help with some claims in an article on Analysis

Question

The following is a piece of a paper I am reading and with which I am having some trouble.

Let $V = V_1 + V_2: \Bbb{R}^n \longrightarrow \Bbb{R}$ be such that

(i) $V_1 \in C^\infty(\Bbb{R}^n)$, $V_1 \geq 0$ and $\partial^\alpha V_1(x) \in L^\infty(\Bbb{R}^n)$ for $|\alpha| \geq 2$

(ii) there exists $q \geq n/2, q \geq 1$ such that $V_2 \in L^q(\Bbb{R}^n) + L^\infty(\Bbb{R}^n)$

Define the functional space $$ X = \{v \in H^1(\Bbb{R}^n, \Bbb{C}) \ : \ V_1(x)|v(x)|^2 \in L^1(\Bbb{R}^n) \}, $$
which is a Hilbert space with the inner product $$ (v, w)_X = \text{Re} \int_{\Bbb{R}^n} v \overline w + \nabla v \cdot \overline{\nabla w} + V_1 v \overline w \ dx, \quad v, w \in X. $$

Define the operator $H : X \longrightarrow X^*$ by $$ \langle Hv, w \rangle = \text{Re} \int_{\Bbb{R}^n} \nabla v \cdot \overline{\nabla w} + V(x) v \overline w \ dx, \quad v, w \in X $$ Then $H$ is well-defined by (i) and (ii). In particular, by (ii) and the Hölder and Sobolev inequalities, for every $\varepsilon > 0$ there exists a constant $C_\varepsilon > 0$ such that for every $v \in H^1(\Bbb{R}^n)$ \begin{equation} (1) \quad \quad \left|\int_{\Bbb{R}^n} V_2(x) |v(x)|^2 \ dx \right| \leq ||V_2||_{L^q + L^\infty}\left(C_\varepsilon ||v||_{L^2}^2 + \varepsilon ||\nabla v||_{L^2}^2 \right) \end{equation}

My questions are all small, I believe, and all related, so I make them all here in a single post:

1. How to prove that $\text{Re}\int_{\Bbb{R}^n} V_1v \overline w \ dx < \infty$ for all $v, w \in X$?

2. What does it mean $H$ is well-defined? Is it that $\langle Hv, w \rangle < \infty$ for all $v, w$? If so, how to prove this?

3. How to prove inequality (1)?

Thanks in advance and kind regards.

EDIT

Regarding 1., is the following argument correct? $$ V_1|v|^2 \in L^1 \implies \sqrt{V_1}|v| \in L^2. $$ Then $$ \left|\int V_1 v \overline w \right| \leq \int \sqrt{V_1}|v| \sqrt{V_1}|w| < \infty $$ by Hölder's inequality.

Answer 1

As for 1., your argument is correct.

2. To be well-defined means that both integrals in the definition make sense. Indeed, the first one makes sense by Cauchy-Schwartz and by $u, v \in H^1$, and the second integral makes sense by item 1..

3. It follows from Gagliardo-Nirenberg inequality, which states that, for $2 < q < 2^*$, there exist $a, b \in (0,1)$ such that $a+b = 1$ and $$ \|u\|_p \lesssim \|u\|_2^a\|\nabla u\|_2^b.$$

Now, writing $V_2 = f_\infty+f_q$, we have $$ \left|\int_{\Bbb{R}^n} V_2(x) |v(x)|^2 \ dx \right| \lesssim \|f_\infty\|_\infty\|v\|_2^2 + \|f_q\|_q\|v\|_{2q'}^{2}.$$

Assuming $q > n/2$, $q \geq 1$, we have $2 \leq 2q' < 2^*$. The case $q' = 1$ is straightforward, so assume $2 < 2q' < 2^*$. In this case, by Gagliardo-Nirenberg and Young,

$$ \|v\|_{2q'}^{2} \lesssim (\varepsilon^{-\frac{b}{a}}\|v\|_2^{2})^a(\varepsilon\|\nabla v\|_2^{2})^b \lesssim C_\varepsilon \|v\|_2^2 + \varepsilon\|\nabla v\|_2^2. $$

The inequality does not seem to hold for $q = n/2$, since this would give $a=0$.