Regularity of General Maximal and Minimal Functions

In this paper, our object of investigation is the endpoint regularity of the following general maximal operator, ℳ_Φ f(x)=sup _r,s≥ 0 r+s>0Φ (r+s)∫ _x-r^x+s|f(y)|dy, and minimal operator, m_Φ f(x)=inf _r,s≥ 0 r+s>0Φ (r+s)∫ _x-r^x+s|f(y)|dy, where Φ (t):(0,∞ )→ (0,∞ ) is a non-increasing continuous function and satisfies B_q:=sup _t>0tΦ (t)^q<∞ for some q≥ 1 . We prove that if f:ℝ→ℝ is a function of bounded variation, then max{Var_q(ℳ_Φ f),Var_q(m_Φ f)}≤ (8B_q)^1/qVar(f). Here, Var_q(f) denotes the q-variation of f and Var_q(f)=Var(f) when q=1 . Similar results are proved for the discrete versions of the above operators.