A counterexample for \(L^ 1\)-estimates for parabolic differential equations (Q686839)

Summary: We show that the Dini (1) continuity of the coefficients of a linear parabolic differential operator in non-divergence form is in some sense the weakest condition such that the solutions of the corresponding initial value problem satisfy an \(L^ 1\)-estimate. Here a function is called Dini \((\alpha)\) continuous for a positive number \(\alpha\) if the modulus of continuity \(\omega\) of the function satisfies \(\int_{0+} (\omega^{1/\alpha} (\tau)/\tau)d\tau<\infty\). In particular, we improve a counterexample of Il'in which shows that an \(L^ 1\)-estimate cannot hold in general if only Dini \((\alpha)\) continuity with \(\alpha<1/4\) is assumed.