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Suppose $f_n$ is sequence of functions such that $f_n\rightarrow f$ in $L^2(\mathbb{R})$. Also suppose $\|f_n\|_p\leqslant C \|f\|_p$, where constant $C$ is independent of $n$. From this data, can we conclude that $f_n\rightarrow f$ in $L^p(\mathbb{R})$? Assume $1<p<\infty$.

Rakesh Balhara
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1 Answers1

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No, you can't. For $p \in (1,\infty)$, $p \ne 2$ you can always achieve $g_n \to 0$ in $L^2(\mathbb{R})$, but $\|g_n\|_{L^p(\mathbb R)} = 1$. For this, it is sufficient to consider $$g_n = \chi_{(0,\alpha_n)} \, \beta_n$$ for suitably chosen scalars $\alpha_n$ and $\beta_n$. Finally, pick your favorite $f \in L^2(\mathbb R) \cap L^p(\mathbb R) \setminus \{0\}$ and consider $f_n = g_n + f$.

gerw
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  • I guess u misunderstood the problem, whatever u are saying is okay but not pertinent to this problem. We are given $f_n\rightarrow f$ in $L^2$ and $L^p$ boundedness. Did u use these both assumptions? – Rakesh Balhara Jan 16 '17 at 14:32
  • u took $f=0$, then $|f|_p = 0$, so its not $L^p$ bounded. – Rakesh Balhara Jan 16 '17 at 14:43
  • @RakeshBalhara This can be easily fixed. Take any $f\in L^2\cap L^p$ with $|f|_p = 1$ and $f_n$ as above. Then $f_n+f\to f$ in $L^2$, $|f_n+f|_p\leq |f_n|_p+|f|_p = 1+|f|_p = 2 = 2|f|_p$. Obviously $f_n+f\not\to f$ in $L^p$. – Lukas Betz Jan 16 '17 at 15:25
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    I edited my answer to obtain $f \ne 0$. Finally, you should bear in mind that your assertion holds for $C = 1$, see http://math.stackexchange.com/questions/163209/weak-convergence-in-lp-plus-convergence-of-norm-implies-strong-convergence. – gerw Jan 17 '17 at 18:12