**5. Conclusions**

As a conclusion to this chapter, we have given necessary and sufficient conditions for the generalized Calderón and Hilbert operators to be bounded from weighted Lesbesgue spaces into suitable weighted *BMO* and Lipschitz spaces. Then, we have obtained results on the boundedness of these operators from *L*<sup>∞</sup> into *BMO*, even in the unweighted case for the Hilbert operator. The class of weights involved are close to the doubling and reverse Hölder conditions related to the Muckenhoupt's classes.

The study of the weighted boundedness for integral operators on function spaces, like the one we develop in this chapter, is one of the main research fields in harmonic analysis. In particular, it has had a profound influence in partial differential equations, several complex variables, and number theory. Evidence of such success and

importance is the pioneering work of leading mathematicians Bourgain, Zygmund, Calderón, Muckenhoupt, Wheeden, C. Fefferman, Stein, Ricci, Tao and so on.
