Abstract
We give a simple proof of the Sawyer type characterization of the two weight estimate for positive dyadic operators (also known as the bilinear embedding theorem).
Supported by the National Science Foundation under the grant DMS-0800876.
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Notes
- 1.
The author is also familiar with a manuscript by F. Nazarov, dated back to the same time as [2], where this result was proved for all p ∈ (1, ∞), again using the Bellman function method. However, this manuscript was never published.
- 2.
Recall that \(\mathcal{L}\subset \mathcal{D}\) is the collection of cubes from Proposition 2.2. However, the construction works for arbitrary \(\mathcal{L}\subset \mathcal{D}\).
- 3.
We are not going to specify what does “reasonable” means, mentioning only that the L p estimates of the maximal function, 1 < p < ∞ are the examples of such estimates.
References
Lacey, M.T., Sawyer, E.T., Uriarte-Tuero, I.: Two weight Inequalities for discrete positive operators (2009). arXiv:0911.3437 [math.CA]
Nazarov, F., Treil, S., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Am. Math. Soc. 12(4), 909–928 (1999)
Sawyer, E.T.: A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75(1), 1–11 (1982)
Sawyer, E.T.: A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Am. Math. Soc. 308(2), 533–545 (1988)
Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge/New York (1991)
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Treil, S. (2015). A Remark on Two Weight Estimates for Positive Dyadic Operators. In: Gröchenig, K., Lyubarskii, Y., Seip, K. (eds) Operator-Related Function Theory and Time-Frequency Analysis. Abel Symposia, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-08557-9_8
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DOI: https://doi.org/10.1007/978-3-319-08557-9_8
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