Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators
Xilin Zhou, Ziyi He, Dachun Yang
Abstract
Abstract Let (𝒳, d , μ ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω . Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:msubsup><m:mrow><m:mi>H</m:mi></m:mrow><m:mo>*</m:mo><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:msubsup><m:mrow><m:mo>(</m:mo><m:mi>𝒳</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> H_*^{p,q}\left( \mathcal{X} \right) with the optimal range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>p</m:mi><m:mo>∈</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mfrac><m:mi>ω</m:mi><m:mrow><m:mi>ω</m:mi><m:mo>+</m:mo><m:mi>η</m:mi></m:mrow></m:mfrac><m:mo>,</m:mo><m:mo>∞</m:mo></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> p \in \left( {{\omega \over {\omega + \eta }},\infty } \right) and q ∈ (0, ∞]. When and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>p</m:mi><m:mo>∈</m:mo><m:mo stretchy="false">(</m:mo><m:mfrac><m:mi>ω</m:mi><m:mrow><m:mi>ω</m:mi><m:mo>+</m:mo><m:mi>η</m:mi></m:mrow></m:mfrac><m:mo>,</m:mo><m:mn>1</m:mn><m:mo stretchy="false">]</m:mo></m:mrow></m:math> p \in ({\omega \over {\omega + \eta }},1] q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.