Oscillation and variation for Riesz transform in setting of Bessel operators on H¹ and BMO

Abstract

Let \lambda >\mathrm{0} and let the Bessel operator \mathrm{\Delta }\lambda ：=-{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}}{{\mathrm{d}x}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{2}\lambda }{x}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}} defined on {{\displaystyle ℝ}}_{+}:=(\mathrm{0},\infty ). We show that the oscillation and \rho-variation operators of the Riesz transform {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} associated with {\mathrm{\Delta }}_{\lambda } are bounded on BMO({{\displaystyle ℝ}}_{+},{{\displaystyle dm}}_{\lambda }), where \rho >\mathrm{2} and {{\displaystyle \mathrm{d}m}}_{\lambda }=x^{\mathrm{2}\lambda }\mathrm{d}x. Moreover, we construct a {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-atom as a counterexample to show that the oscillation and \rho-variation operators of {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} are not bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }). Finally, we prove that the oscillation and the {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-variation operators for the smooth truncations associated with Bessel operators {{\displaystyle \tilde{R}}}_{{\mathrm{\Delta }}_{\lambda }} are bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }).

Key words： Oscillation operator; variation operator; Bessel operator

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