Tohoku Mathematical Journal
2001
June
SECOND SERIES VOL. 53, NO. 2

Tohoku Math. J.
53 (2001), 265-284

Title WEAK TYPE $(1, 1)$ ESTIMATES FOR MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS

Dedicated to Professor Kôzô Yabuta on his sixtieth birthday
Author Dashan Fan and Shuichi Sato

(Received July 23, 1999, revised May 24, 2000)
Abstract. We consider Marcinkiewicz integrals arising from rough kernels satisfying the $L\log L$ condition on the unit $(n-1)$-sphere and prove the weak type (1, 1) estimates. We also prove the weighted weak type (1, 1) estimates with certain $A_1$-weights. In this case the $L\log L$-condition is replaced by the $L^q$-condition with $q>1$.

2000 Mathematics Subject Classification. Primary 42B25.
Key words and phrases. Littlewood-Paley functions, Marcinkiewicz integrals, weak type estimates.

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