Abstract. We study one-sided nonlocal equations of the formon the real line. Notice that to compute this nonlocal operator of order 0 < α < 1 at a point x 0 we need to know the values of u(x) to the right of x 0 , that is, for x ≥ x 0 . We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli-Silvestre and StingaTorrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.
Abstract. In this note we give a simple proof of the characterization of the weights for which the one-sided Hardy-Littlewood maximal functions apply I-PiV) into weak-LP{U) and a direct proof of the characterization of the weights for which the one-sided Hardy-Littlewood maximal functions apply LP(W) into LP(W).
Abstract. In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in (0, ∞) then the one-sided Ap condition, A − p , is a sufficient condition for the singular integral to be bounded in L p (w), 1 < p < ∞, or from L 1 (wdx) into weak-L 1 (wdx) if p = 1. This one-sided Ap condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in (0, ∞). The two-sided version of this result is also obtained: Muckenhoupt's Ap condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in (−∞, 0) or in (0, ∞).
We discuss several characterizations of the A ∞ class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual A ∞ weights.
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