Hölder continuity in time of heat semigroup

Question

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that $$ \|\ell\|_{\infty} + \sup_{\substack{x, y \in {\bR}^d \\ x \neq y}} \frac{| \ell (x)-\ell (y)|}{|x-y|^\alpha} + \int_{\bR^d} |x| \ell (x) \diff x \le c. $$

Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e., $$ p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \exp \left ( -\frac{|x|^2}{4 t} \right ). $$

We denote by $*$ the convolution operation and let $\ell_t := \ell * p_t$.

Is there a constant $c_1>0$ (depending on $d, \alpha, c$) such that we have for $s, t\in [0, 1]$: $$ \int_{\bR^d} (1 + |x|) |\ell_t (x) - \ell_s (x)| \diff x \le c_1 |t-s|^{\frac{\alpha}{2}} \, ? $$

Thank you for your elaboration.

Answer 1

No, such a bound is not possible.

We will show a counterexample for $d = 1$. For a given $T \in \mathbb{Z}_{+}$, consider the following function $\ell^{(T)} : \mathbb{R} \to \mathbb{R}_{\geq 0}$: on the interval $x\in [-1,1]$ define $\ell(x) := (1 - |x|)$, on the interval $x\in[2\pi T, 2\pi(T+1)$ define $\ell(x) := (1 + \sin(x T))/T$, on $x \in [2\pi T - 1, 2 \pi T]$ take $\ell(x)$ to linearly interpolate between $0$ and $1/T$, similarly on the interval $x\in [2\pi (T+1), 2\pi (T+1) + 1]$ and finally $\ell(x) = 0$ elsewhere.

Here is the plot of the proposed function (not to scale).

Clearly $\int \ell(x) \mathrm{d} x \in [1, 10]$, similarly $\int |x| \ell(x) \mathrm{d}(x) \in [0.5, 10]$, and $\ell$ is $1$-Lipschitz.

Now, note that for small enough $t$, as long as $T \gg 1/t$, the difference $\ell_t(X) - \ell(X)$ is very close $\sin(xT)/T$ for $x \in [2 \pi T, 2\pi (T+1)]$, and close to zero everywhere else. This is because $\sin(T x)$ is an eigenfunction of a convolution with a Gaussian kernel $p_t$: if $f_T(x) := \sin(T x)$ then $f_T * p_t = \exp(-t^2 T^2) f_T$. Hence $\int |x| |\ell_t^{(T)}(x) - \ell^{(T)}(x)| \mathrm{d}{x} \approx \int_{2 \pi T}^{2\pi(T+1)} |x| |\sin(Tx)|/|T| \mathrm{d} x \geq \Omega(1)$.

For arbitrarily small $t$, we can chose large enough $T \gg 1/t$, such that the integral $\int |x| |\ell^{(T)}_t - \ell^{(T)}_t| \mathrm{d} x\geq C$, where $C$ is a universal constant.