The Stability of Wavelet-Like Expansions in \(A_\infty \) Weighted Spaces
We prove \(L^p\) boundedness in \(A_\infty \) weighted spaces for operators defined by almost-orthogonal expansions indexed over the dyadic cubes. The constituent functions in the almost-orthogonal families satisfy weak decay, smoothness, and cancellation conditions. We prove that these expansions are stable (with respect to the \(L^p\) operator norm) when the constituent functions suffer small dilation and translation errors.
KeywordsLittlewood–Paley theory Almost-orthogonality Weighted norm inequality Bounded mean oscillation Singular integral operators
Mathematics Subject Classification42B25 primary 42B20 secondary
The author is glad to express his deep gratitude to the anonymous referees, whose careful reading spotted many typographical errors, and whose detailed and insightful comments on style and structure greatly improved the paper’s readability. Also, thanks to them, the author has begun to think about this work, and its relation to other results, in ways that had not occurred to him before. Thank you!
- 2.Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics 79. American Mathematical Society, Providence (1991)Google Scholar
- 9.Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)Google Scholar
- 12.Wilson, M.: Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Springer, New York (2007)Google Scholar
- 16.Wilson, M.: The necessity of \(A_\infty \) for translation and scale invariant almost-orthogonality. In: Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2): Celebrating Cora Sadosky’s Life. AWM-Springer, New York pp. 435–460 (2017)Google Scholar