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Let us denote by $Pr([a,b])$ the set of functions $f:[a,b]\to\mathbb R$, which have a primitive, i.e. an everywhere differentiable function $F:[a,b]\to\mathbb R$ such that $F'(x)=f(x)$ for all $x\in[a,b]$ (Limits at the boundaries are considered to be one-sided).

Now, suppose a function $g:[a,b]\to\mathbb R$ has the following property: For each $f\in Pr([a,b])$ we have $f\cdot g\in Pr([a,b])$. I'd like to show (or disprove), that $g\in L^\infty([a,b])$ (i.e. that $g$ is essentially bounded).

I was able to show that $g$ has the following properties:

  • $g\in L^p([a,b])$ for each $p\in[1,\infty)$.
  • $p\circ g\in Pr([a,b])$ for each polynomial $p$, in particular, $g\in Pr([a,b])$.

The property $g\in Pr([a,b])$ alone is, of course, too weak to guarantee $g\in L^\infty([a,b])$, since there are well-known examples of derivatives which are essentially unbounded.

Any help is highly appreciated. Thanks in advance!

sranthrop
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  • The intervals you are talking about can be unbounded intervals. Here are some examples: https://math.stackexchange.com/questions/257584/can-the-graph-of-a-bounded-function-ever-have-an-unbounded-derivative. – sranthrop Apr 14 '17 at 17:22
  • To what version of the Mean Value Theorem of Bourbaki are you referring? Can you give me an exact reference please? – sranthrop Apr 14 '17 at 18:43
  • @WillM. One well known example is $f(x)=x^2\sin (1/x^2)$ on $[0,1].$ This $f$ is differentiable on $[0,1],$ but $f'([0,1])= \mathbb R.$ – zhw. Apr 14 '17 at 23:32
  • It suffices to prove that if $g\notin L^\infty$, then there exist pairwise disjoint intervals $I_n=[a_n,b_n]$ with $a_n<b_n$, such that $g(x)>n$ for almost all $x\in I_n$. – san Apr 15 '17 at 18:22

1 Answers1

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Let us prove that if $g$ has the "multiplier property", then it is essentially bounded. Clearly, $g^2$ also has the "multiplier property", and $g$ is bounded if and only if $g^2$ is bounded, so with no loss of generality we may assume that $g \ge 0$. Suppose, contrary to our claim, that $g$ is not essentially bounded. We will prove that there is a function $f$ such that $f \in Pr([a,b])$, but $f g \notin Pr([a,b])$.

Step 1. Almost every point of $[a, b]$ is a Lebesgue point of $g$. Hence, for each $n = 1, 2, \ldots$ there is a Lebesgue point $x_n$ of $g$ such that $g(x_n) \ge n$. By passing to a subsequence if necessary, we may assume that $x_n$ is monotone – and thus it converges to some limit $x_0$ – and that $|x_{n+1} - x_0| < \tfrac{1}{n+1} |x_n - x_0|$. With no loss of generality we may assume that $[a, b]$ contains $0$, $x_n$ is decreasing and $x_0 = 0$; thus, $x_{n+1} < \tfrac{1}{n+1} x_n$.

Step 2. Choose $\varepsilon_n > 0$ so that $\varepsilon_n < \tfrac{1}{4} x_n$, and furthermore $$ \tag{1} \operatorname{Leb} \{x \in (x_n + \varepsilon_n, x_n + 3 \varepsilon_n) : g(x) \ge \tfrac{1}{2} g(x_n) \} > \varepsilon_n $$ (this is possible, because $x_n$ is a Lebesgue point of $g$). Next, choose $\alpha_n > 0$ in such a way that $$ \tag{2} \frac{1}{x_k} \sum_{n = k}^\infty \alpha_n \varepsilon_n g(x_n) = 4 $$ (that is, $\alpha_n = 4 (x_n - x_{n+1}) / (\varepsilon_n g(x_n))$). Then clearly $$ \tag{3} \frac{1}{x_k} \sum_{n = k}^\infty \alpha_n \varepsilon_n \le \frac{4}{\inf \{g(x_n) : n \ge k\}} \le \frac{4}{k} \, . $$

Step 3. Define $f$ to be a function which is nonnegative, equal to $0$ outside the union of $(x_n, x_n + 4 \varepsilon_n)$, smooth except at $x_0$, and such that $$ 0 \le f(x) \le \alpha_n $$ when $x \in (x_n, x_n + 4 \varepsilon_n)$, and $$ f(x) = \alpha_n $$ when $x \in (x_n + \varepsilon_n, x_n + 3 \varepsilon_n)$.

Step 4. Let $F$ be the indefinite Lebesgue integral of $f$. Clearly, $F'(x) = f(x)$ for all $x$ except perhaps $x = 0$. However $F'_-(0) = 0$, and if $$ x_k \le x \le x_{k-1} ,$$ then, by (3), $$ 0 \le \frac{F(x) - F(0)}{x - 0} \le \frac{1}{x_k} \sum_{n = k}^\infty \int_{x_n}^{x_n + 4 \varepsilon_n} f(x) dx \le \frac{1}{x_k} \sum_{n = k}^\infty 4 \alpha_n \varepsilon_n \le \frac{16}{k} \, .$$ Therefore, $F'_+(0) = 0$, and hence $F'(0) = 0 = f(0)$. Therefore, $f \in Pr([a, b])$.

Step 5. On the other hand, let $H$ be the indefinite Lebesgue integral of $f g$. We claim that $H$ is not differentiable at $x = 0$. By (1) we have $$ \int_{x_n + \varepsilon_n}^{x_n + 3 \varepsilon_n} f(x) g(x) dx \ge \frac{\alpha_n \varepsilon_n g(x_n)}{2} \, , $$ so that, by (2), $$ \frac{H(x_k + 4 \varepsilon_k) - H(0)}{(x_k + 4 \varepsilon_k) - 0} \ge \frac{1}{2 x_k} \sum_{n = k}^\infty \frac{\alpha_n \varepsilon_n g(x_n)}{2} = 1. $$ Therefore, $$ \limsup_{x \to 0^+} \frac{H(x) - H(0)}{x - 0} \ge 1 . $$ If $a < 0$, then $F'_-(0) = 0$, and we are done.

Step 6. The case $a = 0$ requires an additional calculation. In addition to (1) we need to assume that $$ \int_{x_n}^{x_n + 4 \varepsilon_n} g(x) dx \le 8 \varepsilon_n g(x_n) $$ (again this is possible because $x_n$ is a Lebesgue point of $g$). Then $$ \int_{x_n}^{x_n + 4 \varepsilon_n} f(x) g(x) dx \le 8 \alpha_n \varepsilon_n g(x_n) \, , $$ so that by (2), $$ \frac{H(x_{k-1}) - H(0)}{x_{k-1} - 0} \le \frac{1}{x_{k-1}} \sum_{n = k}^\infty 8 \alpha_n \varepsilon_n g(x_n) = \frac{32 x_k}{x_{k-1}} \le \frac{32}{k} \, . $$ Therefore, $$ \liminf_{x \to 0^+} \frac{H(x) - H(0)}{x - 0} \le 0 . $$ It follows that $F'_+(0)$ does not exist.

Mateusz Kwaśnicki
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  • This seems to be awesome. Thank you! I will work myself through it asap! – sranthrop Apr 16 '17 at 22:21
  • Great job. There are some minor errors. I think we should have $\alpha_n = 4 (x_n - x_{n+1}) / (\varepsilon_n g(x_n))$. Moreover I don't understand (3). I get $$ \frac{1}{x_k} \sum_{n = k}^\infty \alpha_n \varepsilon_n \le \frac{4}{k}. $$ Of course, the whole argument is still valid. – san Apr 17 '17 at 17:48
  • @san: Thanks for the corrections, I just edited the answer to include them. – Mateusz Kwaśnicki Apr 17 '17 at 20:01
  • I don't understand the part where we take the Lebesgue integrals. What if you have a derivative that is not Lebesgue integrable? I.e. if $H$ isn't defined then it wouldn't suffice to show its derivative isn't $fg$ in order to conclude that $fg$ isn't a derivative. – Jonas Meyer Apr 17 '17 at 21:49
  • when you use (3) you obtain $\le \frac{16}k$ instead of $\le \frac 1k$. – san Apr 17 '17 at 21:51
  • @san: Thanks again! – Mateusz Kwaśnicki Apr 17 '17 at 22:03
  • @JonasMeyer: If $F' \ge 0$ exists everywhere, then it is Lebesgue integrable. One reason for this is that $F'$ is always integrable in Henstock–Kurzweil sense, and H–K integral agrees with the Lebesgue integral for nonnegative integrands. A more direct argument is: take the indefinite Lebesgue integral $F_a$ of $\min(F', a)$, and note that $F - F_a$ is absolutely continuous with nonnegative derivative – thus non-decreasing. Use monotone convergence to deduce that $F'$ is integrable. Conversely, if $F'$ is integrable, then $F$ is the indefinite Lebesgue integral of $F'$. – Mateusz Kwaśnicki Apr 17 '17 at 22:14
  • @JonasMeyer: In my more pedestrian point of view. If $F'=f$, then $F$ is continuous on the compact interval . How could it possibly NOT be the Lebesgue integral of $f$? (módulo a constant) – san Apr 18 '17 at 00:54
  • @san it is certainly possible for a function to not be Lebesgue integrable and yet be the derivative of a function. For instance, $F(x)=x^2\sin(\frac1x)$ is differentiable everywhere on $[0,1]$ but its derivative is not integrable. Mateusz Kwasnicki's point was that if you assume $F'\ge0$ then it is integrable. – Jason Apr 18 '17 at 02:18
  • @san: What is the relevance of continuity of F? The potential problem would be where f is not integrable. (I think Jason meant $x^2\sin( 1/x^2)$.). I didn't understand that it wouldn't be a problem given the positivity of f and g in the answer. Thanks, Mateusz, for elaborating. – Jonas Meyer Apr 18 '17 at 02:34
  • Thanks again to all of you for this fruitful discussion. @MateuszKwaśnicki: Can you please explain where (1) comes from exactly? Moreover, shouldn't we have $4\varepsilon_n\leq x_{n-1}-x_n$ to ensure that the intervals $[x_n,x_n+4\varepsilon_n]$ are disjoint (instead of $4\varepsilon_n\leq x_n$)? – sranthrop Apr 18 '17 at 11:41
  • We have $\varepsilon_n < x_n/4$ and $x_n < x_{n-1}/n$, so that $x_n + 4 \varepsilon_n \le 2 x_n \le x_{n-1}$. Property (1) follows from the definition of a Lebesgue point: for $\varepsilon$ small enough, $\frac{1}{8 \varepsilon} \int_{x_n-4\varepsilon}^{x_n+4\varepsilon}|g(x) - g(x_n)|dx<g(x_n)/16$, and so the set of $x$ in this interval such that $|g(x) - g(x_n)| \ge g(x_n)/2$ has Lebesgue measure at most $\varepsilon$. – Mateusz Kwaśnicki Apr 18 '17 at 11:54
  • Got it. Thank you very much! – sranthrop Apr 18 '17 at 14:57