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Function Spaces with Respect to the Gaussian Measure

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Gaussian Harmonic Analysis

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Abstract

One of the main goals of functional spaces is to interpret and quantify the smoothness of functions. In this chapter, we discuss the analogs of classical functional spaces with respect to the Gaussian measure. We see that almost all classical spaces with respect to the Lebesgue measure have an analog for the Gaussian measure; nevertheless, we see that in some cases, for instance, Hardy spaces, the analogs to classical spaces are still incomplete and/or imperfect. On the other hand, most of the time, even if the spaces look similar, most of the proofs are different, mainly because the Gaussian measure is not invariant by translation, which implies the need for completely different techniques.

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Notes

  1. 1.

    In [231], the spaces are defined as completions of \(C_{0} ^{\infty }(\mathbb {R}^d)\). This unfortunate mistake was pointed out in [232]. These spaces, just like other Hardy spaces associated with an operator L, can only be defined on the range of L (where the reproducing formula holds in a L 1 sense). In other situations, this is only a minor technical hindrance. For the Ornstein–Uhlenbeck operator, however, this is critical because of the change of spectrum at p = 1.

  2. 2.

    Actually, Theorem 4.43 gives a slightly stronger inequality involving \(\Upsilon ^{\ast }_\gamma (1,a'),\) the “average” non-tangential maximal function.

  3. 3.

    Also, we obtain the same space with an equivalent norm if instead of \(\mathcal {P}_a,\) we consider \(\mathcal {P}_a\) the admissible cubes of parameter a, i.e., the cubes Q with sides parallel to the axes , with a center at c q and a side length l q ≤ am(cQ).

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Authors and Affiliations

  1. Mathematics and Actuarial Sciences, Roosevelt University, Chicago, IL, USA

    Wilfredo Urbina-Romero

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Urbina-Romero, W. (2019). Function Spaces with Respect to the Gaussian Measure. In: Gaussian Harmonic Analysis. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-05597-4_7

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