The Hardy Space $$H^1$$ H 1 in the Rational Dunkl Setting

Abstract

This paper is perhaps the first attempt at a study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama's approach, we characterize H 1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H 1 . These results are proved here in the one-dimensional case and in the product case.

Hörmander's multiplier theorem for the Dunkl transform

“…This happens e.g. in the case of the product system of roots or radial multipliers (see [4] and [7]). So in order to overcome the lack of knowledge about the Dunkl translation on the L p (dw)-spaces for a general system of roots we use the only information we have, that is, the boundedness of τ x on L 2 (dw) together with very important observation about supports of translations of L 2 (dw)-functions, which is stated in the following our next main result.…”

On Hardy and BMO Spaces for Grushin Operator

“…The estimates (3.3)-(3.5) are consequences of (1.4), (2.21), (2.13), and Corollary 2.5 (see, e.g., [1,Appendix 3]). Now we define the Hardy spaces…”

“…The upper Gaussian estimates (2.13) imply that M L is bounded on L p (X) for 1 < p ≤ ∞ and of weak-type (1,1). We define the Hardy space…”

Hardy’s Inequality for Laguerre Expansions of Hermite Type

“…Namely, consider a measure space (X, µ), where X is one of the domains: R d , (0, ∞) d , d ≥ 1, or (0, π), d = 1, and µ is the corresponding Lebesgue measure. For a suitable orthonormal basis {ϕ n } n∈N d in L 2 (X, µ), the following inequality was studied (1) n∈N d | f, ϕ n | (n 1 + . .…”

“…, n d ), the symbol ·, · denotes the inner product in L 2 (X, µ), and H 1 (X, µ) is an appropriate Hardy space. The main difficulty lies not only in establishing (1), but also in finding the smallest admissible exponent E, for which such inequality holds. Kanjin's research was inspired by the well known Hardy inequality, which states that (see [6])…”