I am reading the paper A Zvonkin's transformation for stochastic differential equations with singular drift and applications where the authors introduce the following function spaces:
We fix $\theta \in [0,2]$ and $p, q \in [1,\infty]$.
Let $H^{\theta, p} := (\mathbb{1}-\Delta)^{-\frac{\theta}{2}}(L^p)$ be the usual Bessel potential space with norm $$ \|f\|_{\theta, p} := \|(\mathbb{1}-\Delta)^{\frac{\theta}{2}} f\|_p . $$
We denote by $\|\cdot\|_{W_q^{\theta, p}(t, T)}$ the norm of $$ W_q^{\theta, p}(t, T) := L^q ([t, T], H^{\theta, p}). $$
We fix $\chi \in C_c^{\infty} (\mathbb{R}^d)$ with $\mathbb{1}_{\{|x| \leq 1\}} \leq \chi \leq \mathbb{1}_{\{|x| \leq 2\}}$. We define for $r>0$ and $x, z \in \mathbb{R}^d$: $$ \chi_r^z(x) := \chi \bigg (\frac{x-z}{r}\bigg). $$
We denote by $\tilde{H}^{\theta, p}$ the localized $H^{\theta, p}$-space: $$ \tilde{H}^{\theta, p}:=\left\{f \in H_{\mathrm{l o c}}^{\theta, p} (\mathbb{R}^d) ; \|f\|_{\tilde{H}^{\theta, p}}:=\sup _z \|\chi_r^z f \|_{\theta, p}<\infty\right\}. $$
The localized space of $W_q^{\theta, p}(t, T)$ is $$ \tilde{W}_q^{\theta, p} (t, T) := \left\{f \in L^q ([t, T], H_{\mathrm{l o c}}^{\theta, p}) ; \|f\|_{\tilde{W}_q^{\theta, p}(t, T)}:=\sup _{z \in \mathbb{R}^d} \|\chi_r^z f\|_{W_q^{\theta, p}(t, T)}<\infty\right\}. $$
We denote by $\nabla, \nabla^2$ the gradient and the Hessian (in weak sense) w.r.t. the spatial variable. Now we fix $p, q \in (1, \infty)$. Is there a constant $c >0$ such that $$ \begin{align} \sup_{t \in [0, T]} \{ \|f(t, \cdot)\|_\infty + \|\nabla f(t, \cdot)\|_\infty \} &\le c \|f\|_{\tilde{W}_\infty^{\frac{3}{2}, \infty} (0, T)}, \\ \|\nabla^2 f\|_{\tilde{W}_q^{0, p} (0, T)} &\le c \|f\|_{\tilde{W}_q^{2, p} (0, T)} \end{align} $$ ?
Thank you so much for your elaboration!