A note on the sunouchi operator with respect to the walsh—kaczmarz system

Abstract

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H ^p to L ^p for all 0<p≤1.. For the proof we consider a multiplier operator and prove its (H ^p H ^p)—boundedness for 0<p≤1. Since the multiplier is obviously bounded from L ² to L ², a theorem on interpolation of operators can be applied to show that our multiplier is of weak type (1,1) and of type (q q) for all 1<q<∞. The same statements follow also for the Sunouchi operator.

Keywords:

• Walsh—Kaczmarz functions
• Hardy spaces
• quasi—local operators

AMS:

• 42C10
• 42C20
• 43A75
• 42B30