**2. Calderón weights and** *L<sup>p</sup>***-weighted inequalities**

A function *ω* defined on *<sup>n</sup>* is called a *weight* if it is locally integrable and positive almost everywhere. For a measurable set *E* ⊂ *<sup>n</sup>*, ∣*E*∣ denote its Lebesgue measure, *<sup>ω</sup>*ð Þ¼ *<sup>E</sup>* <sup>Ð</sup> *<sup>E</sup>ω*, and *<sup>E</sup><sup>c</sup>* the complement of *<sup>E</sup>* in *<sup>n</sup>*. Given a ball *<sup>B</sup>*, *tB* is the ball with the

same center as *<sup>B</sup>* and with radius *<sup>t</sup>* times as long, and *<sup>f</sup> <sup>B</sup>* <sup>¼</sup> <sup>1</sup> ∣*B*∣ Ð *<sup>B</sup>f*. As usual, *χ<sup>E</sup>* denotes the characteristic function of *E* and *B x*ð Þ ,*r* denotes a ball centered at *x* with radius *r*. Also, *C* denotes a positive constant.

Let *ω* be a weight in *<sup>n</sup>* and 1≤*p*< ∞. A Lebesgue measurable function *f* belongs to *Lp*ð Þ *<sup>ω</sup>* if

$$\|f\|\_{L^{p}(\alpha)} = \left(\int\_{\mathbb{R}^{n}} |f|^{p} \, d\nu\right)^{1/p} < \infty.$$

We say that an oprator *<sup>T</sup>* is a bounded operator on *<sup>L</sup>p*ð Þ *<sup>ω</sup>* if

$$\|Tf\|\_{L^{p}(\alpha)} \leq C \|f\|\_{L^{p}(\alpha)}, \qquad \text{for all} f \in L^{p}(\alpha).$$

Given 1<*p* < ∞, it is said that *ω* is a Calderón weight of class C*p*, that is *ω*∈ C*p*, if the Calderón operator *<sup>S</sup>* is bounded on *Lp*ð Þ *<sup>ω</sup>* (see [5]) or, equivalently, if *<sup>P</sup>* and *<sup>Q</sup>* are both bounded on *<sup>L</sup><sup>p</sup>*ð Þ *<sup>ω</sup>* (see also [6]). It is well known that the class <sup>C</sup>*<sup>p</sup>* for *<sup>p</sup>* <sup>&</sup>gt;1 is given by the conditions

$$\begin{aligned} M\_p &: \quad \left(\int\_{[0,\infty]} \boldsymbol{\alpha}(t) dt\right)^{1/p} \left(\int\_{[\boldsymbol{x},\infty)} \frac{\boldsymbol{\alpha}^{1-p'}(t)}{t^{p'}} dt\right)^{1/p'} \leq \mathbb{C} \quad \text{for all} \ \boldsymbol{x} > 0;\\ M^p &: \quad \left(\left(\int\_{[\boldsymbol{x},\infty)} \frac{\boldsymbol{\alpha}(t)}{t^p} dt\right)^{1/p} \left(\int\_{[0,\boldsymbol{x}]} \boldsymbol{\alpha}^{1-p'}(t) dt\right)^{1/p'} \leq \mathbb{C} \quad \text{for all} \ \boldsymbol{x} > 0. \end{aligned}$$

The Calderón operator plays an important role in the theory of real interpolation and such theory related to Calderón weights is developed in [5]. On the other hand, in [7], the authors considered a maximal operator *N* on 0, ð Þ ∞ associated to the basis of open sets of the form 0, ð Þ *b* , given by

$$N\!f(x) = \sup\_{b>x} \frac{1}{b} \int\_{[0,b]} |f(t)|dt$$

for measurable functions *f*. Then, for nonnegative functions *f*, we have

$$P(\boldsymbol{\pi}) \le \mathcal{N}\mathcal{f}(\boldsymbol{\pi}) \le \mathcal{S}\mathcal{f}(\boldsymbol{\pi}) \qquad \text{for all } \boldsymbol{\pi} > \mathbf{0}.$$

The classes of weights *<sup>ω</sup>* associated to the boundedness of *<sup>N</sup>* on *<sup>L</sup><sup>p</sup>*ð Þ *<sup>ω</sup>* are those that satisfy the Muckenhoupt-*Ap* condition, 1≤*p* < ∞, only for the sets of the form ð Þ 0, *b* . These classes are denoted by *Ap*,0 and defined as follows:

$$\begin{aligned} A\_{1,0}: \quad &No(\mathbf{x}) \leq Co(\mathbf{x}) \quad \text{for almost all } \mathbf{x} > \mathbf{0};\\ A\_{p,0}: \quad & \left(\frac{1}{\mathfrak{X}} \int\_{[0,\mathbf{x}]} \boldsymbol{\omega} \right) \left(\frac{1}{\mathfrak{X}} \int\_{[0,\mathbf{x}]} \boldsymbol{\omega}^{1-p'} \right)^{p-1} \leq \mathbf{C} \quad \text{for all } \mathbf{x} > \mathbf{0}, \text{where} \\ &\mathbf{C} \leq C \quad \text{for all } \mathbf{x} > \mathbf{0}, \text{where} \end{aligned}$$

Then, in [7] is proved that *<sup>N</sup>* and *<sup>S</sup>* are bounded operators on *<sup>L</sup><sup>p</sup>*ð Þ *<sup>ω</sup>* if and only if *ω*∈ *Ap*,0 for 1<*p*< ∞. This result implies, in particular, that the classes of weights *Cp* and *Ap*,0 coincide for 1<*p*< ∞.

*A Brief Look at the Calderón and Hilbert Operators DOI: http://dx.doi.org/10.5772/intechopen.106027*

Taking into account these results it is natural to wonder for the action of the Calderón and Hilbert operators over suitable spaces such as *BMO* or Lipschitz spaces. Also, another interesting question is: which are, in these cases, the Calderón weights in order to obtain weighted inequalities between these spaces?

These problems were treated for instance in the case of the fractional integral operator in [8, 9], which have been the main motivation for the article [10] and for the development of the following sections.
