Weighted norm inequalities for the geometric maximal operator

• Autores: C.J. Neugebauer, David Cruz-Uribe
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 42, Nº 1, 1998, págs. 239-263
• Idioma: inglés
• DOI: 10.5565/publmat_42198_13
• Títulos paralelos:
  • Desigualdades de norma ponderada para el operador geométrico máximo
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• Resumen

  • We consider two closely related but distinct operators, $$ \align 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}.

    \endalign $$ 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.