Elona Agora: Weak-type boundedness of classical operators in weighted Lorentz spaces

Abstract: Throughout this talk we will discuss the  boundedness of the Hardy-Littlewood maximal operator, M, and Hilbert transform, H, on weighted Lorentz spaces. These spaces were defined by Lorentz in the 50's and generalize the weighted Lebesgue spaces and the classical Lorentz spaces. On  weighted Lebesgue spaces,  the boundedness of M and H  was characterized by the so-called Ap Muckenhoupt class of weights, applying  techniques from the Calderón-Zygmund theory.  On the other hand, the boundedness of M and H on the classical Lorentz spaces was characterized using tools from the theory of rearrangement invariant spaces. Our aim is to unify and extend these results through the weighted Lorentz spaces. The results are based on joint works with Jorge Antezana, María Jesús Carro, and Javier Soria (see [1], [2], [3]).

[1] E. Agora,  J. Antezana, and M. J. Carro, The complete solution to the weak-type boundedness of Hardy-Littlewood maximal operator on weighted Lorentz spaces, (2016) To appear in J. Fourier Anal. Appl.

[2] E. Agora, J. Antezana, M. J. Carro and J. Soria, Lorentz-Shimogaki and Boyd Theorems for weighted Lorentz spaces, London Math. Soc. 89  (2014) 321-336.

[3] E. Agora, M. J. Carro and J. Soria, Complete characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces, J. Fourier Anal. Appl. 19 (2013) 712-730.