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.
Starting from a slight modification of the dyadic sets introduced by M. Christ in [A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601-628] on a space of homogeneous type (X, d, ), an MRA type structure and a Haar system H controlled by the quasi distance d, can be constructed in this general setting in such a way that H is an orthonormal basis for L 2 (d ). This paper is devoted to explore under which conditions on the measure , the system H is also an unconditional basis for the Lebesgue spaces L p (d ). As a consequence, we obtain a characterization of these spaces in terms of the H-coefficients.
Abstract. In this paper we study inequalities with weights for fractional operators T α given by convolution with a kernel K α which is supposed to satisfy some size condition and a fractional Hörmander type condition. As it is done for singular integrals, the conditions on the kernel have been generalized from the scale of Lebesgue spaces to that of Orlicz spaces. Our fractional operators include as particular cases the classical fractional integral I α , fractional integrals associated to an homogeneous function and fractional integrals given by a Fourier multiplier.Mathematics subject classification (2010): 42B20, 42B25.
We extend the results by Jones and Rosenblatt about the series of the differences of differentiation operators along lacunary sequences to BMO and to the setting of weighted L pspaces. We use a different approach which allows to establish that the one-sided Sawyer A p weights are the natural ones to study the boundedness and convergence of that series in weighted spaces.
Abstract. We characterize the class of weights related to the boundedness of maximal operators associated to a Young function η in the context of variable Lebesgue spaces. Fractional versions of these results are also obtained by means of a weighted Hedberg type inequality. These results are new even in the classical Lebesgue spaces. We also deal with Wiener's type inequalities for the mentioned operators in the variable context. As applications of the strong type results for the maximal operators, we derive weighted boundedness properties for a large class of operators controlled by them.
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