Some remarks on multilinear maps and interpolation

Abstract

A multilinear version of the Boyd interpolation theorem is proved in the context of quasi-normed rearrangement-invariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integrals.

Some Bilinear Estimates

“…Then Theorem D follows by complex bilinear interpolations as in the work of [9], see also the work of [7]. Now return to the case F α ( f , g) (x).…”

Restricted convolution inequalities, multilinear operators and applications

“…For example, if the multiplier is the Fourier transform of the Lebesgue measure on a smooth hypersurface, the derivatives satisfy the same decay estimates as the multiplier, and no better. Another example that is not covered by this result is the case of multilinear fractional integration, handled in [19] using different methods; see also [12].…”

“…We also obtain a maximal variant of the restricted convolution estimate and use it to to obtain multilinear variants of Stein's spherical maximal theorem [24]. Multilinear operators, especially convolutions, have been widely studied; see, e.g., [19,12,13,30,16]. We focus here on showing that a variety of apparently quite different multilinear results can be viewed through the unifying framework of restricted convolution.…”

Translation-Invariant Bilinear Operators with Positive Kernels

“…, as proved independently by Kenig and Stein [8] and Grafakos and Kalton [5]. The bilinear fractional integrals are also operators of the form (1) associated with the singular measures µ α = δ 0 (y + z)|y| −n+α on R n × R n , where 0 < α < n, and they map L p × L q → L r , when 1/p + 1/q = α/n + 1/r.…”