In this talk, I will talk about generalized Schauder type estimates for linear parabolic equations in divergence, non-divergence, and double-divergence form.
We assume that the principal coefficients and certain lower order term coefficients satisfy Dini mean oscillation condition, that is the mean oscillation of the coefficients satisfy the Dini condition; $$extract_itex$$\int_0^1 \omega(t)/t dt <\infty$$/extract_itex$$.
We show that solutions to parabolic equations in divergence form, non-divergence form, and double-divergence form satisfy local and global $$extract_itex$$C^{1/2,1}_{t,x}$$/extract_itex$$, $$extract_itex$$C^{1,2}_{t,x}$$/extract_itex$$, and $$extract_itex$$C^{0,0}_{t,x}$$/extract_itex$$ estimates.