Weak-Type Lower Bounds for High-Dimensional Hardy–Littlewood Maximal Operators on Certain Measures via Averaging Operators

Abstract

AbstractConsider $$\mathbb {R}^d$$ R d with the euclidean distance, and let $$0<\alpha <1$$ 0 < α < 1 . We study the behavior of the averaging operators given by the radial density $$d\mu (x)=|x|^{-\alpha d}dx$$ d μ ( x ) = | x | - α d d x . When $$1\le p<\infty $$ 1 ≤ p < ∞ is such that $$(1-2\alpha )p<1-\alpha $$ ( 1 - 2 α ) p < 1 - α , we show that the weak (p, p) bounds grow exponentially with the dimension d. As a consequence, the corresponding results follow for the centered Hardy–Littlewood maximal operator. The lower bounds obtained here are new for averaging operators, and when $$0<\alpha \le 1/2$$ 0 < α ≤ 1 / 2 and $$p>1$$ p > 1 , they are also new for the maximal operator.