**1. Introduction**

The Calderón and Hilbert operators are among the most relevant operators in harmonic analysis, arising from Hilbert's double series theorem which is one of the simplest and most beautiful in the theory of double series of positive terms. It was proved by Hilbert, in the course of his investigations in the theory of integral equations, that the series P *m*,*n*∈ *am an am*þ*an* , where *an* ≥0 for all *n* ∈ , is convergent whenever P *<sup>n</sup>*<sup>∈</sup> *a*<sup>2</sup> *<sup>n</sup>* is convergent.

Other proofs of Hilbert's double series theorem and generalizations in different directions were studied and published over time by influential mathematicians such as H. Weyl, F. Wiener, J. Schur, Fejér and F. Riesz, Pólya and Szegö, Francis and Littlewood, G.H. Hardy and M. Riesz, among others.

In [1, 2], G.H. Hardy observed that Hilbert's theorem stated above is an immediate corollary of another theorem which has interest in itself. This theorem is as follows: If

*an* ≥0 for all *n*∈ and P *<sup>n</sup>*<sup>∈</sup> *a*<sup>2</sup> *<sup>n</sup>* is convergent, then P *n*∈ 1 *n* P*<sup>n</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>a</sup> <sup>j</sup>* � �<sup>2</sup> is also convergent.

The first extension of the just stated Hilbert's and Hardy's results in which we are interested is the following (see [3]): Let 1< *p*< ∞ and *p*<sup>0</sup> ¼ *p=*ð Þ *p* � 1 (i.e. *p*<sup>0</sup> is the conjugate of *p*). If P<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*<sup>a</sup> p <sup>n</sup>* and P<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>*b<sup>p</sup>*<sup>0</sup> *<sup>n</sup>* are convergent, where *an* and *bn* are nonnegative numbers for all *n*∈ , then

$$\sum\_{m=1}^{\infty} \sum\_{n=1}^{\infty} \frac{a\_m b\_n}{m+n} \le \frac{\pi}{\sin(\pi/p)} \left(\sum\_{m=1}^{\infty} a\_m^p \right)^{1/p} \left(\sum\_{n=1}^{\infty} b\_n^{p'} \right)^{1/p'} \text{ and} \\ \sum\_{n \in \mathbb{N}} \left(\frac{1}{n} \sum\_{j=1}^n a\_j \right)^p \le (p')^p \sum\_{n=1}^{\infty} a\_n^p.$$

The constants *<sup>π</sup><sup>=</sup>* sin ð Þ *<sup>π</sup>=<sup>p</sup>* and *<sup>p</sup>*<sup>0</sup> ð Þ*<sup>p</sup>* <sup>¼</sup> ð Þ *<sup>p</sup>=*ð Þ *<sup>p</sup>* � <sup>1</sup> *<sup>p</sup>* are the best possible.

At the same time, the continuous versions of the previous inequalities are the following (see [3, 4]): Let 1<*p* < ∞ and *p*<sup>0</sup> the conjugate of *p*. If Ð ½ Þ 0,<sup>∞</sup> j j *<sup>f</sup> <sup>p</sup>* and <sup>Ð</sup> ½ Þ 0,<sup>∞</sup> j j *<sup>g</sup> <sup>p</sup>*<sup>0</sup> are finite, then

$$\int\_{[0,\infty)} \int\_{[0,\infty)} \frac{|f(\mathbf{x})| |\mathbf{g}(\mathbf{y})|}{\mathbf{x} + \mathbf{y}} d\mathbf{x} d\mathbf{y} \le \frac{\pi}{\sin(\pi/p)} \left( \int\_{[0,\infty)} |f(\mathbf{x})|^p d\mathbf{x} \right)^{1/p} \left( \int\_{[0,\infty)} |\mathbf{g}(\mathbf{x})|^{p'} d\mathbf{x} \right)^{1/p'} $$

and

$$\int\_{\left[0,\infty\right)} \left(\frac{1}{\varkappa} \int\_{\left[0,\varkappa\right]} f(\jmath) d\jmath\right)^p d\varkappa \le \left(\frac{p}{p-1}\right)^p \int\_{\left[0,\varkappa\right)} |f(\varkappa)|^p d\varkappa.$$

Once again, the constants involved are the best possible.

As usual in harmonic analysis, if *<sup>E</sup>* is a measurable subset of *<sup>n</sup>*, then *Lp*ð Þ *<sup>E</sup>* , 1≤*p* < ∞, is the Lebesgue space of all measurable functions *f* such that ∥*f* ∥ *p <sup>L</sup>p*ð Þ *<sup>E</sup>* ¼ Ð *<sup>E</sup>*<sup>j</sup> *f x*ð Þj*<sup>p</sup> dx* is finite. Recall that *Lp*ð Þ *<sup>E</sup>* , <sup>∥</sup> � <sup>∥</sup>*Lp*ð Þ *<sup>E</sup>* � � is a Banach space and in the case *<sup>E</sup>* <sup>¼</sup> *<sup>n</sup>*, it is denoted <sup>∥</sup> � <sup>∥</sup>*<sup>p</sup>* <sup>¼</sup> <sup>∥</sup> � <sup>∥</sup>*Lp*ð Þ *<sup>E</sup>* .

Now, consider the operators *H* and *P* defined by

$$Hf(\mathbf{x}) = \int\_{[0,\infty)} \frac{f(t)}{\mathbf{x} + t} dt \qquad \text{and} \qquad Pf(\mathbf{x}) = \frac{1}{\mathbf{x}} \int\_{[0,\mathbf{x}]} f(t) dt,$$

which naturally arise from the inequalities presented above. Also consider

$$Qf(x) = \int\_{\left[x,\infty\right)} \frac{f(t)}{t} dt$$

being the adjoint operator of *P* and satisfying

$$\int\_{[0,\infty)} (Q^f(\infty))^p d\mathfrak{x} = \int\_{[0,\infty)} \left( \int\_{[\infty,\infty)} \frac{f(t)}{t} dt \right)^p d\mathfrak{x} \leq \mathfrak{C} \int\_{[0,\infty)} (f(\infty))^p d\mathfrak{x},$$

for all *<sup>f</sup>* <sup>∈</sup>*Lp*ð Þ ½ Þ 0, <sup>∞</sup> , 1<sup>&</sup>lt; *<sup>p</sup>*<sup>&</sup>lt; <sup>∞</sup>, where *<sup>C</sup>* is a positive constant (see [4]). Therefore, *<sup>P</sup>* and *<sup>Q</sup>* are bounded operators from *Lp*ð Þ ½ Þ 0, <sup>∞</sup> in itself, that is,

$$\|Pf\|\_{L^{p}([0,\infty))} \leq C\|f\|\_{L^{p}([0,\infty))} \text{ and } \|Q^{f}\|\_{L^{p}([0,\infty))} \leq C\|f\|\_{L^{p}([0,\infty))} \text{ for all } f \in L^{p}([0,\infty)).$$

It is immediate that for nonnegative functions *f*,

$$Hf(\varkappa) \le Pf(\varkappa) + Qf(\varkappa) \le 2Hf(\varkappa) \qquad \text{for all } \varkappa > 0.$$

Consequently *<sup>H</sup>* is a bounded operator on *<sup>L</sup><sup>p</sup>*ð Þ ½ Þ 0, <sup>∞</sup> , that is,

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

$$\|Hf\|\_{L^{p}([0,\infty))} \leq C \|f\|\_{L^{p}([0,\infty))} \qquad \text{for all } f \in L^{p}([0,\infty)).$$

It is well known that *P* is called the *Hardy averaging operator* and *H* is called the *Hilbert operator*. Also, the *Calderón operator S* is defined by *S* ¼ *P* þ *Q*, being then a bounded operator from *<sup>L</sup>p*ð Þ ½ Þ 0, <sup>∞</sup> in itself.

We end this section with some of the first and most important direct applications obtained from Hilbert's and Hardy's inequalities.

Theorem 1.1 Let *<sup>E</sup>* be the interval 0, 1 ð Þ and *<sup>f</sup>* <sup>∈</sup>*L*<sup>2</sup> ð Þ *E* not null in *E*. Then

$$\sum\_{n=0}^{\infty} \left( \int\_E \mathfrak{x}^n f(\mathfrak{x}) d\mathfrak{x} \right)^2 < \pi \int\_E f^2(\mathfrak{x}) d\mathfrak{x}$$

and the constant *π* is the best possible. The integrals Ð *<sup>E</sup>xnf x*ð Þ*dx*, *<sup>n</sup>* <sup>¼</sup> 0, 1, … are called the *moments of f in E* and are important in many theories.

Theorem 1.2 (Carlema's inequalities) Let f g *an* be a sequence of positive numbers and 1< *p*< ∞. Then

$$\sum\_{n=1}^{\infty} \left( \frac{1}{n} \sum\_{k=1}^{n} a\_k^{1/p} \right)^p < \left( \frac{p}{p-1} \right)^p \sum\_{n=1}^{\infty} a\_n \quad \text{and} \quad \sum\_{n=1}^{\infty} \left( \prod\_{k=1}^{n} a\_k \right)^{1/n} < e \sum\_{n=1}^{\infty} a\_n.$$

The constants involved are the best possible.

The corresponding integral version for the second inequality of Carlema's inequality is: If *f* is a positive function belonging to *L*<sup>1</sup> ð Þ ½ Þ 0, ∞ , then

$$\int\_{\left[0,\infty\right)} \exp\left(\frac{1}{\varkappa} \int\_{\left[0,x\right]} \log f(t)dt\right)d\varkappa = \int\_{\left[0,\infty\right)} e^{P(\log f)(\varkappa)}d\varkappa < e\left[\inf\_{\left[0,\infty\right)} f(\varkappa)d\varkappa.\right]$$

where the constant *e* is the best possible.

Theorem 1.3 Let 1<*<sup>p</sup>* <sup>≤</sup>2 and *<sup>p</sup>*<sup>0</sup> the conjugate of *<sup>p</sup>*. If *Lf s*ðÞ¼ <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> *f t*ð Þ*e*�*st dt*, i.e. *Lf* is the Laplace transform of *f*, then

$$\int\_0^\infty Lf(s)^{p'}ds \le \frac{2\pi}{p'} \left(\int\_0^\infty f(s)^p ds\right)^{p'/p} \qquad \text{for all } f \in L^p([0,\infty)).$$

Therefore *<sup>L</sup>* is a bounded operator from *<sup>L</sup><sup>p</sup>*ð Þ ½ Þ 0, <sup>∞</sup> into *<sup>L</sup><sup>p</sup>*<sup>0</sup> ð Þ ½ Þ 0, ∞ , 1< *p*≤2, and <sup>∥</sup>*Lf* <sup>∥</sup>*<sup>p</sup>*<sup>0</sup> <sup>≤</sup> <sup>2</sup>*π=p*<sup>0</sup> ð Þ<sup>1</sup>*=p*<sup>0</sup> ∥*f* ∥*p*.

The number of applications and results that arise from Hilbert's and Hardy's inequalities is by now very large and it would be impossible to give a detailed survey of all of them in a reasonable amount of text. We have simply made a very brief introduction about them in this section.
