Maybe somebody can help me with the following problem:
On p. 57 of this lecture notes, it is claimed that $$ \lvert I_1(t,t',x)\rvert\leq C\lVert g\rVert_{L^\infty}(t'\log t'-t\log t-(t'-t)\log(t'-t))\tag{2.4.39} $$ implies $$ \lvert I_1'(t,t',x)\rvert\leq C\lVert g\rVert_{L^\infty}\lvert t'-t\rvert^\alpha\tag{2.4.40} $$ for all $\alpha\in (0,1)$.
Here, $t'>t$ as mentioned in the beginning of p. 57.
I have no idea how to see this, do you?
On the first sight, I thought that one has $$ t'\log t'-t\log t-(t'-t)\log(t'-t)\leq \lvert t'-t\rvert^\alpha $$ but this is not correct in general! For example, choose $0<t<t'$ and $\alpha\in (0,1)$ such that both $t'-t$ and $1-\alpha$ are very small.
I think one needs further restrictions on $0<t<t'$ or on $\alpha\in (0,1)$ in order to guarantee the inequality and I see two options.
Option 1
Choose $0<t<t'$. I think one can find $\alpha>0$ small enough such that the inequality holds. The $\alpha$ then should depend on the choice of $t$ and $t'$. And I would call this some kind of "local" argument therefore.
Option 2
If we want to have the inequality for different pairs $(t,t')$ with $0<t<t'$ with a common $\alpha\in (0,1)$, I think one needs to restrict the interval from which we can choose these pairs. So the statement could be something like: "If $0<t<t'<C$, then there exists $\alpha\in (0,1)$ such that the inequality holds for all $t,t'\in (0,C)$."
In both options, the condition "for all $\alpha\in (0,1)$ is replaced.
We used a different part of those notes for a class taught by one of the authors. As suspected by commenter zhw., $C$ is used for essentially "some constant that we don't care enough to specify, whose value may (1) change depending on where it appears and (2) depend on parameters of the situation in an immaterial way." In this instance, as you've found, $C$ cannot always have a consistent value across the two lines, and even just considering $(2.4.40)$ there isn't a single value of $C$ that would work for all $\alpha$. But we're just using $(2.4.40)$ to prove the Hölder continuity of $J(t,x)$, so it suffices to have $$ \lvert t' \log t' - t \log t - (t' - t) \log (t' - t) \rvert \leq C_\alpha \lvert t' - t \rvert^\alpha $$ provided that both $C_\alpha$ and the initial $C$ are independent of $t$, $t'$, and $x$.