Oscillation Inequalities on Real and Ergodic H¹ Spaces

Abstract

Let $({{x}_{n}})$ be a sequence and $\rho \geqslant 1$. For fixed sequences ${{n}_{1}} < {{n}_{2}} < {{n}_{3}} < \ldots$, and $M$, define the oscillation operator

$${{\mathcal{O}}_{\rho }}({{x}_{n}}) = {{\left( {\sum\limits_{k = 1}^\infty {\kern 1pt} \mathop {\sup }\limits_{\substack{ {{n}_{k}} \leqslant m < {{n}_{{k + 1}}} \\ m \in M } } {{{\left| {{{x}_{m}} - {{x}_{{{{n}_{k}}}}}} \right|}}^{\rho }}} \right)}^{{1/\rho }}}.$$

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 $({{n}_{k}})$ and $M$ are lacunary. Then we prove the following results for $\rho \geqslant 2$:

1. 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 $f \in {{H}^{1}}(\mathbb{R})$.

2. 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 $f \in {{H}^{1}}(X)$.

3. If ${{[f(x)\log (x)]}^{ + }}$ is integrable, then ${{\mathcal{O}}_{\rho }}({{A}_{n}}f)$ is integrable.