Question

How to do this question? pleases show some steps.

6. Consider a sequence of continuous function fn : [a,b] → R. Suppose there exist constants 72 1 and ß > 0 independent or, p such that e b for any p > 1 and n e N. Show that there exists a constant C depending on independent of n, such that and B, but rb max lfn(x) rela,b for any n e N. [Hint: Results from Example 4.3.5 may be helpful.]

Answer 1

Solution 6: Consider a sequence of continuous function
a. b*
. Suppose there exists

$\gamma >1, \beta>0$ independent of n,p such that

$\left ( \int_{a}^{b}\left | f_{n}(x) \right |^{p\gamma}dx \right )^{\frac{1}{\gamma}} \leq p.\beta^{\frac{1}{p}}\left ( \int_{a}^{b}\left | f_{n}(x) \right |^{p} dx\right )^{\frac{p-1}{p}}~~~~~~~~~~~~~~~~~~(1)$

Let us consider in particular, a=0, b=1 and $f_{n}(x)=x^n$.

Then for n=1,
fi(x

for n=2,
f2(z) = 12

and so on.

Here all functions are continuous on [0,1].

max In

Now let $\gamma =2>1, , \beta=1>0, p=2>1$ , then from equation (1), we have

$\left ( \int_{0}^{1}\left | f_{n}(x) \right |^{2.2}dx \right )^{\frac{1}{2}} \leq 2.1^{\frac{1}{2}}\left ( \int_{0}^{1}\left | f_{n}(x) \right |^{2} dx\right )^{\frac{2-1}{2}}$

1 1 < 2

$\Rightarrow \left ( \int_{0}^{1}\left | f_{n}(x) \right |^{4}dx \right )^{\frac{1}{2}} \leq 2\left ( \int_{0}^{1}\left | f_{n}(x) \right |^{2} dx\right )^{\frac{1}{2}}< 2\left ( \int_{0}^{1}\left | f_{n}(x) \right |^{2} dx\right )^{\frac{1}{2}}+2$

$\Rightarrow \left ( \int_{0}^{1}\left | f_{n}(x) \right |^{4}dx \right )^{\frac{1}{2}} < 2\left ( \int_{0}^{1}\left ( f_{n}(x) \right )^{2} dx\right )^{\frac{1}{2}}+2$

$\Rightarrow \left ( \int_{0}^{1}\left | f_{n}(x) \right |^{4}dx \right )^{\frac{1}{2}} < 2\left [ \left ( \int_{0}^{1}\left ( f_{n}(x) \right )^{2} dx\right )^{\frac{1}{2}}+1 \right ]$

Since

max In

1 .1

Therefore there exists a constant C depending on $\gamma, \beta$ but independent of n, such that

$\max_{x\n[0,1]}|f_{n}(x)|$$\leq \left ( \int_{0}^{1}\left | f_{n}(x) \right |^{4}dx \right )^{\frac{1}{2}} \leq C\left [ \left ( \int_{0}^{1}\left ( f_{n}(x) \right )^{2} dx\right )^{\frac{1}{2}}+1 \right ]$

Thus there exists a constant C depending on $\gamma, \beta$ but independent of n, such that

$\max_{x\n[a,b]}|f_{n}(x)|$$\leq C\left [ \left ( \int_{0}^{1}\left ( f_{n}(x) \right )^{2} dx\right )^{\frac{1}{2}}+1 \right ]$(Proved)