Sharp Weighted Estimates for Multilinear Commutators

By C. P'erez, Rodrigo Trujillo-González. 2002

270 citations recorded by the source corpus.

Multilinear commutators with vector symbol b→ =(b1,…,bm) defined by Tb→(f)(x)=∫Rn[ ∏j=1m(bj(x)−bj(y)) ]K(x,y)f(y)dy are considered, where K is a Calderón–Zygmund kernel. The following a priori estimates are proved for w ∈ A∞. For 0 < p < ∞, there exists a constant C such that ‖ T˙b→(f) ‖Lp(w)⩽C‖ b→ ‖‖ ML(logL)1/r(f) ‖Lp(w) and supt>01Φ(1t)w({ y∈Rn:|Tb→f(y)|>t })⩽Csupt>01Φ(1t)w({ y∈Rn:ML(logL)1/r(‖ b→ ‖f)(y)>t }), where ‖b→‖=∏j=1m‖bj‖oscexpLrj, Φ(t)=tlog1/r(e+t),1r=1r1+ċ+1rm, and ML(log L)α is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1dc42719d14959fe68cf87a5ac87fec48da54a83

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. P. Auscher
4. J. M. Martell
5. Xing Fu
6. S. Hofmann
7. Guoen Hu
8. Yan Meng

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. P. Auscher
4. J. M. Martell
5. Xing Fu
6. Guoen Hu
7. Yan Meng
8. S. Hofmann

Citations in the corpus

• Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators (2006, background): “7.2, Part (b), in place of those in Theorem 5.2. Details are left to the reader.36 PASCAL AUSCHER AND JOSE MAR´ ´IA MARTELL Similar results can be proved for the multilinear commutators considered in [PT] (see also [AM1]) which are defined by replacing (b(x) − b)k in Tk b by Qk j=1(bj(x) − b) with bj”
• BOUNDEDNESS OF MULTILINEAR COMMUTATORS OF CALDERÓN-ZYGMUND OPERATORS ON ORLICZ SPACES OVER NON-HOMOGENEOUS SPACES (2012, methodology): “To this end, we first recall the following Orlicz type function space OscexpLr (μ) (see, for example, Pérez and Trujillo-González [19] for Euclidean spaces, and [7] for nondoubling measures).”
• Multilinear Commutators of Singular Integrals with Non Doubling Measures (2005, methodology): “for t> 0. Thus by Lemma 2.2 in [ 9 ], we see that for any t0 ,t 1, ··· ,t i > 0,”
• GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, unknown): “To this end, we first recall the following Orlicz type space OscexpLr(μ) of functions (see, for example, Pérez and Trujillo-González [29] for Euclidean spaces and [14] for non-doubling measures).”