Revistes Catalanes amb Accés Obert (RACO)

Weighted norm inequalities for the geometric maximal operator

D. Cruz-Uribe, SFO, C. J. Neugebauer


We consider two closely related but distinct operators,
M_0f(x)&= \sup_{I\ni x}\exp\left(\frac{1}{|I|}\int_I\log|f|\,dy\right) \quad\text{and}\\
M_0^*f(x) &= \lim_{r\rightarrow0} \sup_{I\ni x}\left(\frac{1}{|I|}\int_I|f|^r\,dy\right)^{1/r}.
We give sufficient conditions for the two operators to be equal and show that these conditions are sharp. We also prove two-weight, weighted norm inequalities for both operators using our earlier results about weighted norm inequalities for the minimal operator:
\text{\mgran{m}} f(x) = \inf_{I \ni x} \frac{1}{|I|}\int_I |f|\,dy.
This extends the work of X. Shi; H. Wei, S. Xianliang and S. Qiyu; X. Yin and B. Muckenhoupt; and C. Sbordone and I. Wik.

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