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The ρ-variation of the heat semigroup in the Hermitian setting: behaviour in L

  • J. J. Betancor (a1), R. Crescimbeni (a2) and J. L. Torrea (a3)
Abstract

Let , ρ > 2, be the ρ-variation of the heat semigroup associated to the harmonic oscillator H = ½(−Δ + |x|2). We show that if fL (ℝ), the (f)(x) < ∞, a.e. x ∈ ℝ. However, we find a function GL (ℝ), such that (G)(x) ∉ L (ℝ). We also analyse the local behaviour in L of the operator . We find that its growth is smaller than that of a standard singular integral operator. As a by-product of our work we obtain an L (ℝ) function F, such that the square function

a.e. x ∈ ℝ, where is the classical Poisson kernal in ℝ.

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References
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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