One-sided limit of q norm on $[0,1]$

Question

I'm trying to show that for $f \in L^q([0,1])$ for all $q$ with $1≤q≤p<\infty$, we have $$\lim_{q \to p^-}||f||_q=||f||_p$$

It's easy to show that $$\lim_{q \to p^-}||f||_q≤||f||_p$$ But the other direction is proving difficult. I've tried to use an epsilon argument similar to what you use to show $$\lim_{p \to \infty}||f||_p=||f||_\infty$$but it hasn't worked out.

Answer 1

Suppose first that $f\in L^p$. Let $(q_n)$ be a sequence of numbers smaller than $p$ converging to $p$. Then using the dominated convergence theorem, $\lVert f\rVert_{q_n}^{q_n}\to\lVert f\rVert_{p}^{p}$ (bound $\lvert f\rvert^{q_n}$ by $\lvert f\rvert^p\mathbf{1}_{\{\lvert f\rvert\geqslant 1\}}+\mathbf{1}_{\{\lvert f\rvert\lt 1\}}$).

If $f$ does not belong to $L^p$, let $f_N:=\lvert f\rvert\mathbf{1}_{\{\lvert f\rvert\leqslant N\}}$. Then for all $N\geqslant 1$, $$ \liminf_{q\to p^-}\lVert f\rVert_q\geqslant \liminf_{q\to p^-}\lVert f_N\rVert_q =\lVert f_N\rVert_p,$$ where we used what we established in the first paragraph. By monotone convergence, $\lVert f_N\rVert_p\to +\infty$ as $N$ goes to infinity.

Answer 2

WLOG, $f\ge 0.$ We are letting $q\to p^-,$ where $p\in (0,\infty)$ is fixed. We are assuming $\int_0^1f^p<\infty.$

Observe, as Davide Giraudo pointed out,

$$f^q= f^q\cdot\chi_{\{f\le 1\}} + f^q\cdot \chi_{\{f>1\}}\le 1\cdot\chi_{\{f\le 1\}} + f^p\cdot \chi_{\{f>1\}}$$

for all $q\le p.$ Both summands on the right are in $L^1,$ hence so is their sum. By DCT,

$$\lim_{q\to p^-}\int_0^1 f^q = \lim_{q\to p^-}\int_0^1 \left(f^q\cdot\chi_{\{f\le 1\}} +f^q\cdot \chi_{\{f>1\}}\right)$$ $$ = \int_0^1 \left(f^p\cdot\chi_{\{f\le 1\}} + f^p\cdot \chi_{\{f>1\}}\right) = \int_0^1 f^p.$$

Now

$$\tag1 \|f\|_q = \exp[(1/q)\ln\,(\int_0^1 f^q)].$$

As $q\to p^-,$ use what we have proved, together with continuity, to see the right side of $(1)$ $\to \exp[(1/p)(\ln \int_0^1 f^p)].$ Thus $\|f\|_q\to \|f\|_p$ as desired.

Note we did not have to assume $1\le q.$