On the essential bounded variation of L_p(\mathbb {R},X)-functions

• Autores: Gen Nakamura, Kazuo Hashimoto
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 65, Fasc. 3, 2014, págs. 407-416
• Idioma: inglés
• DOI: 10.1007/s13348-013-0099-y
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• Resumen

  • Let X be a Banach space. Nakamura and Hashimoto in (Proc Japan Acad Ser A 87:77–82, 2011), we showed that for every f\in L_{1}(\mathbb {R}), \begin{aligned} \lim _{h\rightarrow 0}\int \limits _{\mathbb {R}}\left| \frac{f(t+h)-f(t)}{h}\right| \, dt=\hbox {ess} V_{1}(f). \end{aligned} In this paper, we are concerned with the limit \begin{aligned} \lim _{h\rightarrow 0}\int \limits _{\mathbb {R}}\left\| \frac{f(t+h)-f(t)}{h}\right\| ^p\, dt \end{aligned} (*) for f\in L_1^{\hbox {loc}}(\mathbb {R},X). We show that the limit (*) coincides with the essential p-variation of f in the sense of F. Riesz, and we give characterizations of functions with bounded essential p-variation, i.e, \hbox {ess} V_p(f,X) \infty.