My question came from page 392 of of D. Applebaum's book "Levy processes and Stochastic calculus".
I will put it in a nutshell.
Let $\nu$ be a measure on $B_c(0)\setminus\{0\}$ satisfying $\int_{0<|x|<c}|x|^2\,\nu(dx)<\infty$. Function $F:\mathbb R^d\times(B_c(0)-\{0\})\to\mathbb R$ satisfies \begin{equation}\tag 1 \int_{0<|x|<c}|F(y_1,x)-F(y_2,x)|^2\,\nu(dx)<K|y_1-y_2|^2,\quad \forall y_1,y_2\in\mathbb R^d, \end{equation} and \begin{equation}\tag 2 |F(y,x)|\le|\rho(x)||\delta(y)|,\quad \forall y\in\mathbb R^d，x\in B_c(0)\setminus\{0\}, \end{equation} where $\rho:B_c(0)\setminus\{0\}\to\mathbb R$ satisfies $\int_{0<|x|<c}|\rho(x)|^2\,\nu(dx)<\infty$ and $\delta:\mathbb R^d\to\mathbb R^d$ is Lipschitz continuous.
Question: Prove that for each $x\in B_c(0)\setminus\{0\}$, the mapping $y\to F(y,x)$ is continuous.
What I can conclude is from $(1)$ that $$ F(y_n,x)\to F(y,x) \text{ in } L^2(B_c(0)\setminus\{0\},\nu), \quad \text{as } y_n\to y, $$ as a consequence, there is a subsequence $\{y_{n_k}\}$ such that $ F(y_{n_k},x)\to F(y,x)$ for $\nu$-a.s. $x$. But how to get the continuity of $F(\cdot,x)$ for each $x$?
Any comments will be appreciated. Thanks in advance.
Under the given assumptions, continuity of $F(\cdot,x)$ for $\nu$-almost all $x$ is the best what we can expect.
Suppose, for instance, that $\nu$ is absolutely continuous with respect to Lebesgue measure. Define
$$F(y,x) := \begin{cases} yx, & x \neq \frac{c}{2}, \\ x 1_{\mathbb{Q}}(y) & x = \frac{c}{2} \end{cases}.$$
Obviously, $y \mapsto F(y,c/2)$ is not continuous. On the other hand, we have
$$|F(y,x)| \leq |\varrho(x)| \, |\delta(y)|$$
for $\varrho(x) := x$ and $\varrho(y) := \max\{|y|,1\}$. Moreover
$$\begin{align*} \int_{0<|x|<c} |F(y_1,x)-F(y_2,x)|^2 \, \nu(dx) &= \int_{0<|x|<c, x \neq \frac{c}{2}} |F(y_1,x)-F(y_2,x)|^2 \, \nu(dx) \\ &= |y_1-y_2|^2 \int_{0<|x|<c} x^2 \, \nu(dx) \\ &=: K |y_1-y_2|^2 \end{align*}$$ which means that $F$ satisfies all the given assumptions.