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.


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