Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

By P. Auscher, J. M. Martell. 2006

202 citations recorded by the source corpus.

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-? inequality with two parameters and the other uses Calderon?Zygmund decomposition. These results apply well to singular ?non-integral? operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, ?non-integral? that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<8. Pointwise estimates are then replaced by appropriate localized Lp?Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,8) and isolate the right class of weights. In particular, we prove an extrapolation theorem ?a la Rubio de Francia? for such a class and thus vector-valued estimates.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/33321d3f38816f8d3c2060c7f4223ffcd55416c7

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Citations in the corpus

Extrapolation and interpolation in generalized Orlicz spaces (2018, background): “Our final result generalizes the limited range extrapolation theorem of Auscher and Martell [3].”
Extrapolation for multilinear Muckenhoupt classes and applications (2018, background): “There are other versions considering estimates within the smaller classes of weights of the form A p p− ∩ RH(p+ p )′ which are naturally adapted to range p ∈ (p−, p+) (see [1]) or off-diagonal extrapolation results for the classes Ap,q which are natural for estimates from L p to L where p 6= q (see ”
Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “By [Auscher and Martell 2007a, Theorem 3.13], T also satisﬁes some weighted estimates: for every p 2 . p 0 ; q 0 / and every weight !”
Weighted norm inequalities, off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type (2006, unknown): “Although this is not needed in this work, let us mention a reasonable sufficient condition insuring this comparability (see [ AM1 ] for a proof): Assume that there exists e ∈ (0, 1) such that for any ball B ⊂ X ,”
New Abstract Hardy Spaces (2007, methodology): “See also the work of P.Auscher, A.McIntosch and E. Russ [ 7 ], where a similar construction is done on a Riemannian manifold with doubling property, L being”
Extrapolation and weighted norm inequalities in the variable Lebesgue spaces (2014, methodology): “Auscher and Martell [4] proved that there exist constants q − = q − ( L ) < 2 n n +2 < 2 and q + = q + ( L ) > 2 such that if q − p q + and w ∈ A p/q − ∩ RH ( q + /p ) (cid”