Weak type estimates for maximal operators with a cylindric distance function

Abstract

We establish sharp weak-type estimates for the maximal operators T λ * associated with cylindric Riesz means for functions on H p (R 3 ) when 4/5 < p < 1 and λ = 3/ p − 5/2, and when p = 4/5 and λ > 3/ p − 5/2.

Riesz means associated with convex polygons in R2

“…This type of restriction on p has been occurred in [1,2]. In [1] it was observed that this restriction is due to the singularities in the interior of the unit disc {ξ : ρ(ξ ) 1}. Actually if P is a square, then the regularity of the multiplier breaks down along line segments connecting the origin and the vertices of the rectangle so the increase of the power δ does not linearly improve the differentiability of the multiplier.…”

“…It is worthy of noticing that there is restriction on p. This distinguishes Marcinkiewicz-Riesz means from Bochner-Riesz means. This type of restriction on p has been occurred in [1,2]. In [1] it was observed that this restriction is due to the singularities in the interior of the unit disc {ξ : ρ(ξ ) 1}.…”

Riesz means associated with certain product type convex domain

“…As for p < 1, it has been obtained for k = 2 that the maximal cylinder operator is of weak type (p, p) on H p (R 3 ) when 4 5 < p < 1 and δ = 3( 1 p − 1) + 1 2 , or when p = 4 5 and δ > 3( 1 p − 1) + 1 2 in [7]. It has been also shown that the above estimates are sharp.…”

“…However the existence of a disc in the product causes a more interesting feature of the operator due to the non-vanishing curvature of the boundary of the disc. The results in [7,17] are on the multiplier associated with a product of a disc and one interval and those in [12,13] are on the multipliers associated with a product of only intervals. Even though operators of the first type conveys property caused by the curvature, operators of the second type also have interesting mapping property because of the interactions between intervals.…”

Weak type estimates for maximal operators over convex hypersurfaces

“…P. Taylor improved the ranges of p and δ for higher dimensions R n (n 3) in [11]. For the case of p < 1, it is known that the maximal operator associated with the radial cylinder multiplier (1 − max |ξ |, |ξ 3 | ) δ + satisfies sharp weak type (p, p) estimates on H p R 3 when 4/5 < p < 1 and critical index δ = δ(p) := 3(1/p − 1) + 1/2, or p = 4/5 and δ > δ(p) in [3].…”