Note on norm of an m-linear integral-type operator between weighted-type spaces

We find a necessary and sufficient condition for the boundedness of an m -linear integral-type operator between weighted-type spaces of functions, and calculate norm of the operator, complementing some results by L. Grafakos and his collaborators. We also present an inequality which explains a detail in the proof of the boundedness of the linear integral-type operator on L p ( R n ) $L^{p}({\mathbb {R}}^{n})$ space.


Introduction
Throughout this note the set of natural numbers is denoted by N, the set of reals by R, the set of positive reals by R + , the Euclidean n-dimensional space with the norm |x| = ( n j=1 x 2 j ) 1/2 , x = (x 1 , . . . , x n ), by R n , the unit sphere in R n by S, the n -1-dimensional surface measure by dσ (ζ ), σ n = σ (S), the normalized surface measure dσ (ζ )/σ n is denoted by dσ N (ζ ), the open unit ball in R n by B, the open unit ball in R n centered at a and with radius r by B(a, r), the Lebesgue volume measure on R n by dV (x), v n = V (B), the normalized Lebesgue volume measure dV (x)/v n is denoted by dV N (x), whereas L p α ( ) denotes the weighted Lebesgue space on a domain ⊆ R n with the weight w(x) = |x| α , that is, where 1 ≤ p < +∞, α > -n, and dV α (x) = |x| α dV (x) (for α = 0, the space is reduced to the standard L p space on the domain; see, e.g., [1]). If k, l ∈ N are such that k ≤ l, then the notation j = k, l stands for the set of all j ∈ N such that k ≤ j ≤ l.
The following integral-type operator is a basic linear operator which has been studied on many spaces of functions. From the main result in [8] (see also [9]) we have that the operator is bounded on L p (R + ) space when p > 1. This result was later improved in [10] by proving the following formula: for p > α + 1 in nowadays terminology. Although there are many linear operators whose norms can be calculated (see, e.g., [1, and the related references therein), they are, in fact, quite rare since for many more other operators the norms can be only estimated by some quantities. Some of these operators are integral-type ones, a topic of a considerable recent interest (see, e.g., [11, 13-21, 23-25, 28-34] and the related references therein).
Operator (1) was generalized in [35] by introducing the following n-dimensional integral-type operator: for nonnegative locally integrable functions on R n . In [11] it was shown that the norm of the operator H : L p (R n ) → L p (R n ) can be calculated. Namely, the following formula holds: which matches the formula in (2) with α = 0.
Note that H(f )(xζ ) = H(f )(x) for every ζ ∈ S, that is, the function H(f )(x) is radial for every f , and where we have used the change of variables y = |x|z and the fact that V (B(0, |x|)) = |x| n V (B).
In [13] the following m-linear extension of operator (3) was introduced: where m ∈ N, x ∈ R n \ {0}, y 1 , . . . , y m ∈ R n , y j = (y 1 j , . . . , y n j ), j = 1, m, Note is radial for every f , and where we have used the change of variables y j = |x|z j , j = 1, m.
Our main aim here is to complement the results in [13] by calculating the norm of the operator We also explain a detail appearing in the proof of the main result in [11].
The following known formula, which transforms integrals in the Descartes coordinates to the polar ones in R n , will be frequently used in the section that follows: where f is a nonnegative measurable function (see, e.g., [1, pp. 149-150]).

Main results
This section presents and proves our main results in this note.

Boundedness of operator (5) between weighted-type spaces
First we consider operator (3). We consider it separately since its proof explains the first main step in the proof of the general case and does not request complex calculation.
Theorem 1 Let α ∈ (0, n). Then the operator H is bounded on H ∞ α (R n ). Moreover, the following formula holds: Then it is clear that Further, we have for each x = 0, from which it follows that Equalities (10) and (11) imply On the other hand, we have for every f ∈ H ∞ α (R n ), from which it follows that and consequently the boundedness of the operator when α ∈ (0, n). From (12) and (13) the equality in (8) follows.
Remark 1 Note that (10) holds for each α > 0. However, if α ≥ n, then by using formula (7), we have Hence, in this case the operator is not bounded on H ∞ α (R n ). From this and Theorem 1 we obtain the following corollary.
Then the operator where y = (y 1 , y 2 , . . . , y m ). (15) is satisfied, then the following formula for the norm of the operator holds:
Proof Let the family of functions f α be defined in (9). Then clearly relation (10) holds. By using the condition in (14), after some simple calculation it follows that for each x = 0.