a Bound for functions in $L^p$ after convolution with a $G_\lambda$ almost a heat Kernel

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?