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Let $f$ be a non-negative real valued function on $[a,b]$, and let $p:[a,b]\to(1,\infty)$ such that $f^p\in L^1([a,b])$. Let $p_n:[a,b]\to(1,\infty)$ be a (uniformly bounded) sequence of (step-)functions converging to $p$ wrt. the $L^1$-norm, i.e. $||p_n-p||_{L^1([a,b])}\to0$ as $n\to\infty$.

Question: Does \begin{align*} \lim_{n\to\infty}\int_a^bf(x)^{p_n(x)}dx=\int_a^bf(x)^{p(x)}dx \end{align*} hold in general?

If $f$ is bounded, the proof is easy using the Lipschitz continuity of exponential functions. But what if $f$ is unbounded? I neither know if $f^{p_n}$ is in $L^1$, nor I could find a counterexample. Any help, ideas, hints or even counter examples are highly appreciated. Thanks in advance!

sranthrop
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1 Answers1

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The answer is negative: On $[0,1]$ define $p\equiv 2, p_n(x) = 2 +\chi_{(1/n,1/(n-1))}, n= 2,3,\dots$ Then $p_n \to p$ in $L^1.$

Now define

$$f_n(x) = \frac{1}{2^n}\frac{\chi_{(1/n,1/(n-1))}}{\sqrt {x-1/n}|\ln (x-1/n)|}$$

and put $f=\sum f_n.$ Then $\int_0^1 f^2 < \infty.$ But for each $n$

$$\int_0^1 f^{p_n} \ge \int_{1/n}^{1/(n-1)}f_n^3 = \frac{1}{2^{3n}}\int_{1/n}^{1/(n-1)}\frac{1}{(x-1/n)^{3/2}|\ln (x-1/n)|^3 } \ dx = \infty.$$

zhw.
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