**3. The** *n***-dimensional Calderón and Hilbert operators**

For 0 <sup>≤</sup>*<sup>α</sup>* <sup>&</sup>lt;*n*, *<sup>f</sup>* a Lebesgue measurable function and *<sup>x</sup>*<sup>∈</sup> *<sup>n</sup>*, *<sup>x</sup>* 6¼ 0, the general *n*-dimensional Calderón and Hilbert operators are defined by

$$S\_q f(\mathbf{x}) = P\_q f(\mathbf{x}) + Q\_q f(\mathbf{x}) \quad \text{and} \quad H\_q f(\mathbf{x}) = \int\_{\mathbb{R}^n} \frac{f(y)}{(|\mathbf{x}| + |y|)^{n - \alpha}} dy,$$

where *<sup>P</sup>αf x*ð Þ¼ <sup>1</sup> j j *<sup>x</sup> <sup>n</sup>*�*<sup>α</sup>* Ð ∣*y*∣≤∣*x*∣ *f y*ð Þ*dy* and *<sup>Q</sup>αf x*ð Þ¼ <sup>Ð</sup> ∣*y*∣ >∣*x*∣ *f y*ð Þ j j *<sup>y</sup> <sup>n</sup>*�*<sup>α</sup> dy*.

Again, it is immediate that for nonnegative functions *f*, the following pointwise inequalities hold

$$H\_q f(\mathbf{x}) \le \mathbb{S}\_q f(\mathbf{x}) \le 2^{n-a} H\_q f(\mathbf{x}),\tag{1}$$

and consequently, all weighted-*Lp* inequalities obtained for *S* are true for *H* and reciprocally.

In spite of the punctual comparison (1), we will show in Section 4 that the results obtained for *S<sup>α</sup>* and *H<sup>α</sup>* are not analogous when the *BMO<sup>γ</sup>* and Lipschitz spaces are involved.

Both operators *S<sup>α</sup>* and *H<sup>α</sup>* appear in several different contexts and applications, see for instance [4, 11–17].

Next, we introduce the spaces of functions and the classes of weights which appear in our main results.

Recall that a measurable function *f* defined on *E*⊂ *<sup>n</sup>* is said to be *essentially bounded* provided there is some *M* ≥ 0, called an *essential upper bound* for *f*, for which ∣*f x*ð Þ∣ ≤ *M* for almost all *x*∈*E*. As usual, the class of all functions that are essentially bounded on *<sup>E</sup>* is denoted by *<sup>L</sup>*<sup>∞</sup>ð Þ *<sup>E</sup>* and <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup><sup>∞</sup> is the infimum of the essential upper bounds for *<sup>f</sup>* <sup>∈</sup>*L*<sup>∞</sup>ð Þ *<sup>E</sup>* . Then, *<sup>L</sup>*<sup>∞</sup> ð Þ ð Þ *<sup>E</sup>* , <sup>∥</sup> � <sup>∥</sup><sup>∞</sup> is a Banach space.

Now, a Lebesgue measurable function *<sup>f</sup>* belongs to *<sup>L</sup>*<sup>∞</sup>ð Þ *<sup>ω</sup>* if <sup>∥</sup>*fω*∥<sup>∞</sup> <sup>&</sup>lt; <sup>∞</sup>.

Also recall that *L*<sup>1</sup> *loc <sup>n</sup>* ð Þ denotes the space of locally integrable functions *<sup>f</sup>* satisfying that ∥*f χB*∥<sup>1</sup> is finite for every ball *B*⊂ *<sup>n</sup>*.

**Definition 3.2.** Let *ω* be a weight in *<sup>n</sup>* and 0 ≤*γ* <1*=n*. A locally integrable function *f* belongs to *BMO<sup>γ</sup>* ð Þ *ω* if there exists a constant *C* such that for every ball *B*⊂ *<sup>n</sup>*,

$$\frac{1}{\alpha(B)|B|^{\mathcal{I}}} \int\_{B} |f - f\_{B}| \leq C. \tag{2}$$

The seminorm of *f* ∈*BMO<sup>γ</sup>* ð Þ *ω* , ∥*f* ∥*BMO<sup>γ</sup>* ð Þ *<sup>ω</sup>* , is the infimum of all such *C*.

**Definition 3.4.** Let *ω* be a weight in *<sup>n</sup>* and 0≤ *γ* < 1*=n*. A locally integrable function *f* belongs to *BM<sup>γ</sup>* <sup>0</sup>ð Þ *ω* if there exists a constant *C* such that

$$\frac{1}{\alpha(B)|B|^{\mathcal{I}}} \int\_{B} |f| \leq \mathcal{C} \tag{3}$$

for every ball *B*⊂ *<sup>n</sup>* centered at the origin.

The norm of *f* ∈*BM<sup>γ</sup>* <sup>0</sup>ð Þ *ω* , denoted by ∥*f* ∥*BM<sup>γ</sup>* <sup>0</sup>ð Þ *<sup>ω</sup>* , is the infimum of all such *C*. We will denote by *BM*0ð Þ¼ *<sup>ω</sup> BM*<sup>0</sup> <sup>0</sup>ð Þ *ω* .

Observe that with these definitions the space *BMO*0ð Þ *<sup>ω</sup>* is the weighted version of *BMO* introduced by Muckenhoupt and Wheeden in [18]. Also, the family of spaces *BMO<sup>γ</sup>* ð Þ *ω* is contained in the family of weighted Lipschitz spaces I*ω*ð Þ*γ* defined and studied in [8], and *BMO<sup>γ</sup>* ð Þ *ω* for *ω* � 1 is the well known Lipschitz integral space. Furthermore, we note that *<sup>L</sup>*<sup>∞</sup> *<sup>ω</sup>*�<sup>1</sup> ð Þ⊂*BM*0ð Þ *<sup>ω</sup>* <sup>∩</sup> *BMO*ð Þ *<sup>ω</sup>* .

Given *p* >1, it is known that a weight *ω* satisfies the reverse Hölder inequality with exponent *p*, denoted by *ω*∈*RH p*ð Þ, if

$$\left(\frac{1}{|B|}\int\_{B}\alpha^{p}\right)^{1/p} \leq C\frac{1}{|B|}\int\_{B}\alpha\tag{4}$$

for all balls *B*⊂ *<sup>n</sup>* and some constant *C*.

**Definition 3.7.** Given *p* >1, a weight *ω* belongs to *RH*0ð Þ *p* if it satisfies (4) but only for balls centered at the origin.

**Definition 3.8.** A weight *ω* belongs to *D*<sup>0</sup> if it satisfies the doubling condition *<sup>ω</sup>*ð Þ <sup>2</sup>*<sup>B</sup>* <sup>≤</sup>*Cω*ð Þ *<sup>B</sup>* for every ball *<sup>B</sup>* <sup>⊂</sup> *<sup>n</sup>* centered at the origin and some constant *<sup>C</sup>*.

**Definition 3.9.** Let *η*≥ 1, a weight *ω* belongs to *D<sup>η</sup>* if it satisfies the doubling condition

$$\frac{o(2B(\varkappa, |\varkappa| + r))}{|B(\varkappa, |\varkappa| + r)|^\eta} \le C \frac{o(B(\varkappa, r))}{|B(\varkappa, r)|^\eta}$$

every ball *B x*ð Þ ,*<sup>r</sup>* <sup>⊂</sup> *<sup>n</sup>* and some constant *<sup>C</sup>*.

It is immediate that *D<sup>η</sup>* ⊂ *D*<sup>0</sup> for all *η*, and *D<sup>η</sup>* is increasing in *η*. It is well known that each weight in the Muckenhoupt class *A*<sup>∞</sup> is in *RH p*ð Þ ∩ *D<sup>η</sup>* for some *p* and for some *η*, see for instance [19]. On the other hand, there exist weights belonging to *D<sup>η</sup>* for some *η*, such that it is not in *A*∞, see [20].

Also, we observe the following property that we will use along this chapter. If *ω*∈ *D<sup>η</sup>* there exists a constant *C* such that

$$
\alpha o(B) \le \text{Co}\left(B \mid \frac{1}{2}B\right) \tag{5}
$$

for every ball *B*⊂ *<sup>n</sup>* centered at the origin.

**Definition 3.11.** Let 0≤ *α*<*n* and 1< *p*< ∞. A weight *ω* belongs to H0ð Þ *α*, *p* if there exists a constant *C* such that

$$\left(\int\_{B^r} \frac{\alpha^{p'}(y)}{|y|^{(n-a+1)}p'} dy\right)^{1/p'} \le C \frac{\alpha(B)}{|B|^{1+1/p-a/n+1/n}}\tag{6}$$

for every ball *B*⊂ *<sup>n</sup>* centered at the origin.

A weight *ω* belongs to H0ð Þ *α*, ∞ if there exists a constant *C* such that

$$\int\_{B^c} \frac{\alpha(\mathbf{y})}{|\mathbf{y}|^{n-a+1}} d\mathbf{y} \le C \frac{\alpha(B)}{|B|^{1-a/n+1/n}} \tag{7}$$

for every ball *B*⊂ *<sup>n</sup>* centered at the origin.

The classes of weights H0ð Þ *α*, *p* and H0ð Þ *α*, ∞ satisfying (6) and (7) respectively but for all ball *B* ⊂ *<sup>n</sup>*, were introduced and studied in [8].
