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Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that $$ \big\|\sup_{|y|\le R}|f_n(\cdot,y)-f(\cdot,y)|\big\|_{L^p}\to 0\quad as\quad n\to\infty ?? $$

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  • $\begingroup$ Have you tried to approximate $f$ with a linear combination of simple function first, and then to approximate each simple function with a smooth one? $\endgroup$ – Skeeve Mar 25 '19 at 14:00
  • $\begingroup$ I tried to approximate $f$ by functions of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(y)$. But is it true that we can get$$\|\sup_{|y|\le R}|f_n(\cdot,y)-f(\cdot,y)|\|_{L^p}\to 0???.$$ $\endgroup$ – Wenguang Zhao Mar 25 '19 at 15:36
  • $\begingroup$ I think this should be true if the condition $f\in L^p(R^d, X)$ is understood in the sense that $f$ is Bochner integrable with exponent $p$. $\endgroup$ – Skeeve Mar 25 '19 at 16:24
  • $\begingroup$ The conclusion you mentioned seems to need the condition that $X$ is a reflective Banach space, is it right? $\endgroup$ – Wenguang Zhao Mar 25 '19 at 16:55
  • $\begingroup$ I don't think reflexivity is needed here, see e.g. Proposition 2.1.10 in the book Nonlinear Analysis by Leszek Gasinski, Nikolaos S. Papageorgiou $\endgroup$ – Skeeve Mar 25 '19 at 17:04

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