Oscillation inequalities on real and ergodic H1 spaces II

İngilizce (OpenAlex)

Let $$({{x}_{n}})$$ be a sequence and $$\rho \geqslant 1$$ . For two fixed sequences $${{n}_{1}} < {{n}_{2}} < {{n}_{3}} < \ldots $$ , and $$M$$ define the oscillation operator $${{\mathcal{O}}_{\rho }}({{x}_{n}}) = {{\left( {\sum\limits_{k = 1}^\infty \,\mathop {\sup }\limits_{\substack{ {{n}_{k}} \leqslant m < {{n}_{{k + 1}}} \\ m \in M } } {{{\left| {{{x}_{m}} - {{x}_{{{{n}_{k}}}}}} \right|}}^{\rho }}} \right)}^{{1/\rho }}}.$$ Let $$(X,\mathcal{B},\mu ,\tau )$$ be a dynamical system with $$(X,\mathcal{B},\mu )$$ a probability space and $$\tau $$ a measurable, invertible, measure preserving point transformation from $$X$$ to itself. Suppose that the sequences $$({{n}_{k}})$$ is a lacunary, and $$M$$ is any sequence of positive real numbers such that there exists an $$\ell \in \mathbb{R}$$ satisfying $$\# \{ m \in M:{{n}_{k}} \leqslant m < {{n}_{{k + 1}}}\} \leqslant \ell $$ for all $$k \in \mathbb{N}$$ to obtain the above mentioned results, where $$\# $$ denotes cardinality. Then we prove the following results for $$\rho \geqslant 2$$ : (i) Define $${{\phi }_{n}}(x) = \frac{1}{n}{{\chi }_{{[0,n]}}}(x)$$ on $$\mathbb{R}$$ . Then there exists a constant $$C > 0$$ such that $${{\left\| {{{\mathcal{O}}_{\rho }}({{\phi }_{n}} * f)} \right\|}_{{{{L}^{1}}(\mathbb{R})}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(\mathbb{R})}}}$$ for all $$f \in {{H}^{1}}(\mathbb{R})$$ . (ii) Let $${{A}_{n}}f(x) = \frac{1}{n}\sum\limits_{k = 1}^n \,f({{\tau }^{k}}x)$$ be the usual ergodic averages in ergodic theory. Then $${{\left\| {{{\mathcal{O}}_{\rho }}({{A}_{n}}f)} \right\|}_{{{{L}^{1}}(X)}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(X)}}}$$ for all $$f \in {{H}^{1}}(X)$$ . (iii) If $${{[f(x)\log (x)]}^{ + }}$$ is integrable, then $${{\mathcal{O}}_{\rho }}({{A}_{n}}f)$$ is integrable. In the author’s previously published article titled “Oscillation inequalities on real and ergodic $${{H}^{1}}$$ spaces” the above results have been obtained when both $$({{n}_{k}})$$ and $$M$$ are lacunary. Thus the results of this work extents those results to a nonlacunary sequence $$M$$ with a more general growth condition.