We all know that the space of smooth functions on Euclidean space with compact support is dense in the Lp spaces, for p strictly less than infinity. Now my question is: suppose there is a function f which is in Lp space as well as essentially bounded. Is it possible to find a smooth function g with compact support such that it approximates the original function in Lp norm as well as its supremum controlled by the supremum of f, on the support of g?
2026-03-28 10:02:09.1774692129
Boundedness of smooth functions approximating an Lp function
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To be more precise than the comments, use the usual approximation by convolution. This will give you a $C^\infty$ function which is bounded by the $L^\infty$ norm of $f$ and is arbitrarily close to $f$ in $L^p$.
Now truncate by a suitable $C_c^\infty$ bump function $\varphi$ which fulfills $0 \leq \varphi \leq 1$.