The following questiion comes from the article of Stroock & Varadhan (Diffusion processes with continuous coefficients I - 1969 - pg 378 )
We consider the operator $G_\lambda$ $$G_\lambda f(s,x) = \int_s^\infty \int_{\mathbb{R}^d} \frac{1}{\big(2\pi (t-s)\big)^{d/2}} \exp\big\{-\frac{\vert y-x \vert^2}{2(t-s)} \big\}e^{-\lambda(t-s)}f(t,y)\, dy \,dt$$
Define
$$g_\lambda(s,x) = \frac{1}{\big(2\pi (s)\big)^{d/2}}\exp\big\{-\frac{\vert x \vert^2}{2(s)} \big\}e^{-\lambda(t-s)} 1_{s>0}$$
therefore we can view $G_\lambda f(s,x)$ as a convolution (in $\mathbb{R}\times \mathbb{R}^d$) $$g_\lambda * f (s,x) = \int_0^\infty \int_{\mathbb{R}^d} g_\lambda(t-s,y-x) f(t,y) \,dy \, dx$$
One can check that $\vert\vert g_\lambda \vert\vert_{L^q} < \infty$ for q < (d+2)/d (it suffices to integrate in space, one obtains $C_{\lambda,q}\int_0^\infty t^{\phi(q)} e^{-t}\,dt$, now just check when $\phi(\alpha) = - \frac{d}{2} (\alpha-1)> -1$ this yields the result)
Similarly one can see that $\vert\vert \nabla g_\lambda \vert\vert_{L^q} < \infty$ for $q < (d+2)/(d+1)$
Using Hölder's inequality one can see that for $ p : \frac{1}{p} + \frac{1}{q} = 1 $ and $q<(d+2)/d$ $$\sup_{s,x} \big\vert(G_\lambda) f(s,x)\big\vert \leq C_{\lambda, p} \vert\vert f\vert\vert_p$$
Moreover for q < (d+2)/(d+1) Hölder inequality gives us that ($\frac{1}{p} +\frac{1}{q} = 1$) $$ \big\vert G_\lambda f (s,x_1) - G_\lambda(s,x_2) \big\vert \leq \bar{C}_{\lambda, p} \vert\vert f\vert\vert_p \vert x_1-x_2\vert$$
So far I have followed,
Now the authors make two claims:
"Moreover, $G_\lambda f $ has uniformly continuous first derivatives in $x$
and
\begin{multline} \big \vert G_\lambda f (s,x_1 + h) - G_\lambda(s,x_1) - G_\lambda f (s,x_2 + h) + G_\lambda(s,x_2)\big \vert \leq w_0(\vert h\vert) \bar{C}_{\lambda,p}\vert\vert f\vert\vert_p \vert x_1-x_2\vert \end{multline} where $w_0(\vert h\vert) \downarrow 0 $ as $\vert h\vert \downarrow 0$ "
to see
$G_\lambda f $ has uniformly continuous first derivatives
I tried $$ \big \vert \partial_i G_\lambda f (s,x) - \partial_i G_\lambda(s',x') \big \vert = \big \vert \lim_{h \to 0} h^{-1}[G_\lambda f (s,x + h e_i ) - G_\lambda f (s',x' + h e_i ) - (G_\lambda f (s,x) - G_\lambda f (s',x' )] \big \vert \\ =\big \vert \lim_{h \to 0} h^{-1}[g_\lambda *(f - \tau_v f) (s,x + h e_i ) - g_\lambda *(f - \tau_v f)_\lambda(s,x) ] \big \vert \\ \leq \bar{C}_{\lambda,p} \vert \vert f - \tau_v f \vert \vert_p \underset{v \to 0}{\longrightarrow} 0 $$
where $v = (s' -s , x' -x)$ and $\tau_v f (a,b) = f(a + s'-s,b + x'- x)$
To the last claim however I couldn't arrive at any answer, unless
$$\vert \vert f - \tau_v f\vert \vert_p \leq w_{0}(\vert h\vert )\,\vert \vert f\vert \vert_p $$
But I think this is not true.
Any ideas?
We can find that \begin{align*} &\big \vert G_\lambda f (s,x_1 + h) - G_\lambda(s,x_1) - G_\lambda f (s,x_2 + h) + G_\lambda(s,x_2)\big \vert = \vert G_\lambda'(s,\tilde{x}_1)h - G_\lambda'(s,\tilde{x}_2)h\big \vert\\ &\leq h C_{\lambda,q}\|g'_\lambda\|_q \|f - \tau_{\tilde{x}_2 - \tilde{x}_1}f\|_p \end{align*}
And consider the following as a bad typo: