# Can the integral inherit the Lipschitz continuity of its integrand?

Let $$C$$ be the set of continuous functions on $$[0,T]$$ taking values in $$[0,1]$$. Denote $$\|f-g\|_t:=\max_{0\le u\le t}|f(u)-g(u)|$$ for $$f,g \in C$$ and $$t\in [0, T]$$. Let $$\phi: C\times C\times \{(s,t): 0\le s be a measurable function s.t.

$$\Phi(f,g,t):=\int_0^t\phi(f,g,s,t)ds \quad \le \quad \sqrt{t},\quad \forall f,g\in C \mbox{ and } t\in [0,T].$$

If $$\phi(f,g,s,t)\le \|f-g\|_t\big/(t-s)$$ for all $$f,g\in C$$ and $$0\le s, can we prove (or find a counterexample), for some constant $$L>0$$,

$$\Phi(f,g,t)\quad\le\quad L\sqrt{t}\|f-g\|_t,\quad \forall f,g\in C \mbox{ and } t\in[0, T]?$$

PS : It might be useful to give an alternative formulation (according to my context). Indeed, $$\phi$$ appearing above satisfies

$$\begin{eqnarray} \phi(f,g,s,t) &\le& \frac{1}{\sqrt{t-s}},\quad \forall f,g\in C \mbox{ and } 0\le s

Can we deduce, for some $$L>0$$ and $$a \in [1/2,1)$$,

$$\phi(f,g,s,t) \quad \le\quad \frac{L\|f-g\|_t}{(t-s)^a},\quad \forall f,g\in C \mbox{ and } 0\le s

$$\phi(f,g,s,t):=\min\left(\frac {\|f-g\|_t}{t-s},\frac{1}{\sqrt{t-s}}\right).$$
Then for any $$f,g\in C$$ with $$\delta:=\|f-g\|_t \le \sqrt{t}$$, it holds that
$$\Phi(f,g,t)=\int_0^{t-\delta^2}\frac {\|f-g\|_t}{t-s} +\int_{t-\delta^2}^t\frac{1}{\sqrt{t-s}}=\delta\left(\log\left(\frac{t}{\delta^2}\right)+2\right),$$