I am reading the paper Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications by F.O. Porper and S.D. Eidel'man. Below, the cited paper is
- [2] :S.D. Eidel'man, Parabolicheskie sistemy, Nauka, Moscow 1964. MR 29 # 4998. Translation: Parabolic systems, Noordhoff, Groningen, North-Holland, Amsterdam 1969.
I would like to apply Theorem 1.2(2) with $d=0$ in my paper because it does not impose (A2) (which requires the continuity in time of the coefficients). However, I do not know what $(\partial_x+\partial_{\xi})^l$ means and I do not have access to [2].
Could you explain what $(\partial_x+\partial_{\xi})^l$ means? Could you provide some (more recent) references containing results similar to Theorem 1.2(2)?
Thank you so much for your help!
We fix $T \in (0, \infty)$ and let $\mathbb T$ be the finite interval $[0, T]$. Let $a: \mathbb T \times \mathbb R^n \to \mathbb R^n \otimes \mathbb R^n$ and $b:\mathbb T \times \mathbb R^n \to \mathbb R^n$ and $d:\mathbb T \times \mathbb R^n \to \mathbb R$ be measurable such that $a(t, x)$ is a symmetric matrix. We consider the equation $$ \begin{align} 0 &= \mathcal L u \tag{1}\label{1} \\ &:= \partial_t u - \sum_{k, j=1}^n a_{k, j} (t, x) \partial_{x_k} \partial_{x_j} u(t,x) \\ & \quad - \sum_{k=1}^n b_k(t, x) \partial_{x_k} u(t, x) - d(t, x) u(t, x). \end{align} $$
We list the conditions on the coefficients:
(A1) There is a constant $\mu \geqslant 1$ such that $$ \mu^{-1}|\xi|^2 \le \sum_{i, j=1}^n a_{i j}(t, x) \xi_i \xi_j \leqslant \mu|\xi|^2 \quad \text{for all} \quad \xi \in \mathbb R^d . $$
(A2) The coefficients in \eqref{1} are continuous and bounded by a constant $M_0$.
(A3) The coefficients in \eqref{1} satisfy the Dini condition uniformly in $x$, that is, their modulus of continuity $\omega_{t, x}(h) \le \omega(h)$, where $\omega(h)$ satisfies the Dini condition $\int_0^a \frac{\omega(h)}{h} \, \mathrm d h<\infty$.
We call the class of functions satisfying (A3) the Dini class and denote it by $D_1$. If $F(a)=\int_0^a \frac{\omega(h)}{h} \, \mathrm d h$ has the property $\int_0^b \frac{F(a)}{a} \, \mathrm d a < \infty$, then we denote by $D_2$ the subclass of functions from $D_1$ with this additional property. In particular, functions for which $\omega(h)=H h^\alpha$ with $0<\alpha \le 1$, belong to $D_2$; such functions satisfy a Hölder condition in $x$ with constants $H$ and $\alpha$.
Let $Z(t, x; \tau, \xi)$ be the fundemantal solution of the Cauchy problem for \eqref{1}. The following result is established in detail in [2] (English pp. 23-28).
Theorem 1.1. Under the conditions (A1) and (A2) and a Hölder condition in $x$ with constants $H$ and $\alpha$ the Cauchy problem for \eqref{1} has a unique classical fundamental solution $Z(t, x ; \tau, \xi)$ whose derivatives with respect to the basic variables satisfy the inequalities $$ | \partial_t^{m_0} \partial_x^m Z(t, x ; \tau, \xi) | \le C (t-\tau)^{-\left(n+2 m_0+|m|\right) / 2} \exp \left\{-c \frac{|x-\xi|^2}{t-\tau}\right\}, \tag{2}\label{2} $$ where $0<t-\tau \leqslant T, 2 m_0+|m| \le 2$, and the constants $c$ and $C$ depend on $\mu, n, M_0, H, \alpha$ and $T$.
We also need estimates of the derivatives of the fundamental solution that can be obtained under additional smoothness of the coefficients (see [2], Property 5 and Property 6; English pp. 97, 101-102, and [15], Theorem 3).
Theorem 1.2.
- If (A1) holds and the coefficients in \eqref{1} have $k_0+|k|$, $2 k_0+|k| \le r$, derivatives with respect to $t$ and $x$ that are continuous, bounded, and Hölder in $x$, then the fundamental solution has derivatives of order $m_0+|m|, 2 m_0+|m| \le r+2$, with respect to $t$ and $x$, with the estimate \eqref{2}.
- If (A1) holds and the coefficients in \eqref{1} have $l,|l| \le r$, derivatives with respect to $x$ that are continuous, bounded, and Hölder in $x$, then the following estimates hold: $$ \begin{align} &|(\partial_x+\partial_{\xi})^l \partial_t^{m_0} \partial_x^m Z(t, x ; \tau, \xi)| \\ \le & C(t-\tau)^{-\left(n+2 m_0+|m|\right) / 2} \exp \left\{-c \frac{|x-\xi|^2}{t-\tau}\right\}, \quad 2 m_0+|m| \leqslant 2, \quad |l| \leqslant r . \tag{3}\label{3} \end{align} $$
