An elementary proof of the A 2 bound
A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved...
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| Published in | Israel journal of mathematics Vol. 217; no. 1; pp. 181 - 195 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Springer Nature B.V
01.03.2017
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0021-2172 1565-8511 |
| DOI | 10.1007/s11856-017-1442-x |
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| Abstract | A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting. |
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| AbstractList | A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting. |
| Author | Lacey, Michael T. |
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