# On Green's function in the book of A. Friedman

I'm reading the book Partial Differential Equations of Parabolic Type by Friedman, and I have a question on Green's function defined in p.82 (Sec.4/Chap.3). Assume $$b, \sigma: \mathbb R^2 \to \mathbb R$$ are as 'nice' as possible. Let $$G:(x,t;\xi,\tau)\to G(x,t;\xi,\tau)\in\mathbb R$$ be defined as the function s.t. for any continuous function $$f$$ having a compact support, the function

$$u(t,x):=\int_0^\infty G(x,t;\xi,\tau)f(\xi)d\xi$$

satisfies

$$\partial_t u(x,t)= \frac{\sigma^2(t,x)}{2}\partial^{2}_{xx}u(x,t) - b(t,x)\partial_xu(x,t),\quad \forall t>\tau\ge 0, x, \xi>0$$

and $$\lim_{t \searrow \tau} u(x,t)=f(x)$$ for all $$x\ge 0$$ and $$u(0,t)=0$$ for all $$t>0$$. Theorem 16 ensures the wellposedness of $$G$$. Compared to the fundamental solution, is there any reference on the upper bound of $$\partial_x G$$ and $$\partial^{2}_{xx}G$$? Namely, $$\exists \phi_t:\mathbb R^2_+\to\mathbb R_+$$ s.t.

$$|\partial_{x}G(x,t;\xi,0)|+|\partial^{2}_{xx}G(x,t;\xi,0)|\le \phi_t(x;\xi)\quad\mbox{ and } \quad \int_0^\infty \phi_t(x,\xi)dx<\infty?$$

It is known that the fundamental solution admits such an estimation using parametrix. Can we expect a similar result for Green's function?