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Is $(f ast Okay)” in L^1(mathbb R)$ for $f in L^1 cap L^infty(mathbb R)$ and $Okay in BV(mathbb R)$?

Is it workable to infer that $$(f ast Okay)” in L^1(mathbb R)$$ if $f in L^1 cap L^infty(mathbb R)$ and $Okay in BV(mathbb R)$? What I can show is that $(f ast Okay)’ in L^1 cap L^infty$. Is the certain on the second spinoff too undoubted?

If the above doesn’t suffice, is the outcome undoubted with the extra assumption $f in BV(mathbb R)$?

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