**Changmei Tan**

## Integral Inequalities For Littlewood--Paley
Operators

**Abstract:**

Let Tf denote the g-function g(f) or the area integral s(f). We give a
characterization of functions $\Phi$ for which

$$ \Phi(\lambda)\big|\big\{x\in R^n:\;|Tf(x)|>\lambda\big\}\big|\leq C \int_{R^n}
\Phi(|f(x)|)\,dx,$$

$$\int_{R^n} \Phi(|Tf(x)|)\,dx \leq C \int_{R^n} \Phi(|f(x)|)\,dx, $$

where the constant C is independent of f.

**Keywords:**

Littlewood-Paley g-function, area function, Orlicz space.

**MSC 2000:** 42B25