Question on Absolute continuity, bounded variation and Lipschitz continuity

real-analysis

Let $f\colon [a,b]\to\mathbb R$ be an absolutely continuous function whose derivative $f'$ is in $L^p[a,b],\ 1<p<+\infty$. Let $F$ be the total variation function of $f$ given by
$$
F(x)=\sup\sum_{i=1}^n|f(t_i)-f(t_{i-1})|,
$$
where the supremum is taken over all partitions $a=t_0<t_1<\dots<t_n=x$ of $[a,x],\ a\le x\le b$.

Prove that:

  1. $f$ has bounded variation on $[a,b]$, i.e., $F(b)<+\infty$.
  2. for all $x,y\in[a,b],\ x<y$, one has $$
    F(y)-F(x)\le C\cdot |y-x|^\alpha,
    $$
    for some constants $C, \alpha>0$, independent of $x$ and $y$.

For (1) I use the definition of absolutely continuous which gives us $\sum_{i=1}^n|f(t_i)-f(t_{i-1})|<\epsilon$, so if we take supremum it would be finite and by definition of bounded variation if total variation is finite then it call bounded variation. Is my concepts correct?? Please anyone help me.

For number (2) I am having confusion how should I proceed. I would really appreciate if anyone can help me on these two problems.

Best Answer

"...how should I proceed?"

This is one of those problem assignments that merely requires you to assemble a few of the mathematical jigsaw pieces that you presumably have learned. It is not something to go blindly ahead trying to sort out from first principles.

  1. It is completely routine, standard textbook material that every function that is absolutely continuous on an interval $[a,b]$ is also of bounded variation there. It seems bizarre to ask this as part of an assignment. What? You want me to copy the text? Or just cite it?

If you don't have a text handy here is a typical one in reference [1]. See Section 5.7.2 A characterization of absolutely continuous functions.

  1. Another (again standard fact) is that if $f$ is absolutely continuous on $[a,b]$ and $F$ is the variation function (as given) then for any $a \leq x<y\leq b$ we must have $$ F(y)-F(x) = \int_x^y |f'(t)|\,dt .$$

  2. One more fragment that you need and we are done. If $\frac1p+\frac1q=1$ then

\begin{equation*} \int_x^y |f'(t)g(t)| \,dt \leq \left( \int_x^y |f'(t)|^p \,dx \right)^{1/p} \left(\int_x^y |g(t)|^q \,dt \right)^{1/q} \end{equation*}

This is Hölder's inequality, also standard in any textbook presentation where you have encountered (as here evidently) the space $L_p[a,b]$. Again, if one is not handy try Section 13.1.1 Hölder’s inequality in [1].

I think that is maybe enough hints.

REFERENCE.

[1] http://classicalrealanalysis.info/documents/BBT-AlllChapters-Landscape.pdf

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