Weak-type inequalities for Fourier multipliers with applications to the Beurling-Ahlfors transform

作者：Osekowski Adam^*

来源：Journal of the Mathematical Society of Japan, 2014, 66(3): 745-764.

DOI：10.2969/jmsj/06630745

• Fourier multiplier
• singular integral
• Beurling-Ahlfors transform
• martingale
• differential subordination

The paper contains the study of weak-type constants of Fourier multipliers resulting from modulation of the jumps of Levy processes. We exhibit a large class of functions m : R-d -> C, for which the corresponding multipliers T-m satisfy the estimates
parallel to T(m)f parallel to(Lp, infinity(Rd)) <= [1/2 Gamma(2p-1/p-1)]((p-1)/p) parallel to f parallel to(Lp(Rd))
for 1 < p < 2, and
parallel to T(m)f parallel to(Lp, infinity(Rd)) <= [p(p-1)/2](1/p) parallel to f parallel to(Lp(Rd))
for 2 <= p < infinity. The proof rests on a novel duality method and a new sharp inequality for differentially subordinated martingales. We also provide lower bounds for the weak-type constants by constructing appropriate examples for the Beurling-Ahlfors operator on C.

• 出版日期2014-7