Summary
In this paper we consider two nonstandard applications of the Laplace method. The first one is referred to the Inversion Formula of D.V. Widder and E.L. Post, from which a maximal theorem is proved. The second one is a special Calderón–Zygmund partition that gives us a genuine generalization of Natanson’s lemma in this context.
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Notes
- 1.
Natanson’s lemma: Given − ≤ a < b ≤ ∞, a Borel measure μ with support in (a, b) and K(r, x, y) be a Natanson kernel (i.e., K it is monotone increasing in y, for a < y < x, monotone decreasing in y, for b > y > x and \(\int _{a}^{b}K(r,x,y)\mu (dy) \leq M,\) for some M is independent of x and r. Then, for f ∈ L 1(μ), we have \(\vert \int _{a}^{b}K(r,x,y)f(y)\mu (dy)\vert \leq Mf_{\mu }^{{\ast}}(x).\)
References
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Calderón, C.P., Urbina, W.O. (2014). Some Non Standard Applications of the Laplace Method. In: Georgakis, C., Stokolos, A., Urbina, W. (eds) Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics & Statistics, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-319-10545-1_6
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DOI: https://doi.org/10.1007/978-3-319-10545-1_6
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