Abstract
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}({mathbb {R}}^{n}) space.
Highlights
Throughout this note the set of natural numbers is denoted by N, the set of reals byR, the set of positive reals by R+, the Euclidean n-dimensional space with the norm |x| = ( n j=1 x2j )1/2, x = (x1, xn), byRn, the unit sphere in Rn by S, the n
The functional · Hα∞ is a norm on the space, where, as usual, we identify functions which are dV almost everywhere equal
28] and the related references therein), they are, quite rare since for many more other operators the norms can be only estimated by some quantities
Summary
28] and the related references therein), they are, quite rare since for many more other operators the norms can be only estimated by some quantities. For nonnegative locally integrable functions on Rn. In [11] it was shown that the norm of the operator H : Lp(Rn) → Lp(Rn) can be calculated. Fj, j = 1, m, are nonnegative locally integrable functions on Rn, and the norm of the m-linear operator was calculated from the product of weighted Lebesgue spaces m j=1
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