Weighted estimates for multilinear Fourier multipliers

Abstract

Abstract. We prove a Hörmander type multiplier theorem for multilinear Fourier multipiers with multiple weights. We also give weighted estimates for their commutators with vector BM O functions.

A counterexample to weighted estimates for multilinear Fourier multipliers with Sobolev regularity

“…In the case q 1 = · · · = q N = N n/s, this coincides with the results of [1,13] (see (1.5)). However, in the same way as in the proof of Theorem 1.1, we can prove that the estimate T m L p 1 (w1)×···×L p N (wN )→L p (ν w ) sup j∈Z m j W (s 1 ,...,s N ) (R N n ) does not in general hold for w ∈ A P / Q (but this estimate holds for w ∈ N k=1 A p k /q k , see (1.4) and also [3, Therem 6.2] for general case).…”

“…This result can also be obtained from another approach of Hu and Lin [8]. Replacing W (s/N,...,s/N ) by W s , Bui and Duong [1], Li and Sun [13] proved that if w = (w 1 , . .…”

Sparse Domination Theorem for Multilinear Singular Integral Operators with Lr-Hörmander Condition

“…As we have discussed in the above, all these estimates apply to the multilinear Fourier multipliers with symbols satisfying (3.6). Notice that the qualitative result was obtained by the author and Sun in [24].…”

Multilinear Fourier multipliers on variable Lebesgue spaces

“…In [27], Li and Sun got some weighted estimates of T m with multiple weights under the Hörmander conditions in terms of the Sobolev regularity.…”

“…In this paper, we study the weighted estimates of T m with nearly the same conditions as in [27], but on variable exponent Lebesgue spaces.…”