The sharp weighted bound for general Calderón–Zygmund operators

Abstract

For a general Calderón–Zygmund operator $T$ on $\Bbb{R}^N$, it is shown that $$ \Vert{Tf}\Vert{L^2(w)}\leq C(T)\cdot\sup_Q\Big(∫_Q w\cdot ∫_Q w^{-1}\Big)\cdot\Vert{f}\Vert{L^2(w)}$$ for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Pérez–Treil–Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper.
The proof consists of the following elements: (i) a variant of the Nazarov–Treil–Volberg method of random dyadic systems with just one random system and completely without “bad” parts; (ii) a resulting representation of a general Calderón–Zygmund operator as an average of “dyadic shifts;” and (iii) improvements of the Lacey–Petermichl–Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

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      ZBLNUMBER = {1213.42072},
      }
  • [Xiang] Go to document Q. X. Yang, "Fast algorithms for Calderón-Zygmund singular integral operators," Appl. Comput. Harmon. Anal., vol. 3, iss. 2, pp. 120-126, 1996.
    @article {Xiang, MRKEY = {1385048},
      AUTHOR = {Yang, Q. X.},
      TITLE = {Fast algorithms for {C}alderón-{Z}ygmund singular integral operators},
      JOURNAL = {Appl. Comput. Harmon. Anal.},
      FJOURNAL = {Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications},
      VOLUME = {3},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {120--126},
      ISSN = {1063-5203},
      MRCLASS = {42B20 (47G10 65R10)},
      MRNUMBER = {1385048},
      MRREVIEWER = {D. S. Lubinsky},
      DOI = {10.1006/acha.1996.0011},
      ZBLNUMBER = {0859.65052},
      }

Authors

Tuomas P. Hytönen

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