On possible deterioration of smoothness under the operation of convolution (Q5890354)

Let \(\mu\) be a probability measure on the real line. The author asks if there exists a non-negative function \(f\in L^1({\mathbb R})\) with the property that the convolution \(f\ast\mu\) is 'exceptionally irregular' in the sense that (P) ess\( \sup_{x\in I} (f\ast\mu)(x)=\infty\) for every interval \(I\). It is established that the answer is positive if and only if the Hardy-Littlewood maximal function \(M_\mu\) of \(\mu\) satisfies \(\limsup_{|x|\to\infty}M_\mu(x)=\infty\). Moreover, this condition is also sufficient for the existence of a non-negative integrable entire function \(f\) satisfying (P). The author also gives some other conditions which are sufficient for the existence of entire functions \(f\) of finite order \(\rho>1\) satisfying (P).