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Sufficient conditions for one-sided operators. (English) Zbl 0984.42010
The author considers maximal one-sided operators on $$\mathbb R$$ such as $M_{L^p}^+ f(x) := \sup_{r > 0} \Biggl(\int_0^1 |f(x+rt)|^p dt\Biggr)^{1/p}$ or more generally, for a Banach space $$X$$ $M_X^+ f(x) := \sup_{r > 0} \|f(r\cdot) \chi_{[x, x+r]}(r \cdot) \|_X.$ The authors study the question of whether such operators are bounded from one weighted $$L^p$$ space $$L^p(v^p dx)$$ to another $$L^p(w^p dx)$$. Their first result is that if $$M_X^+$$ is bounded on $$L^p(dx)$$ and $$v$$, $$w$$ obey a certain $$A_p$$ weight condition (adapted to $$X$$), then $$M_X^+$$ is also bounded on the weighted spaces. Some necessary conditions for $$M_X^+$$ to be bounded on $$L^p$$ are then given. Weighted estimates are then given for certain one-sided singular integrals (i.e., Calderón-Zygmund operators whose kernel is supported on the half-space $$\{ (x,y): x > y\}$$), in terms of the above maximal operators.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 26A33 Fractional derivatives and integrals
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