Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

By Songbai Wang, Dachun Yang, Wen Yuan, Yangyang Zhang. Journal of Geometric Analysis, 2019

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Let X be a ball quasi-Banach function space on $${\mathbb R}^n$$ R n . In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ W H X ( R n ) under some weak assumptions on the Littlewood–Paley functions. The authors also prove that the real interpolation intermediate space $$(H_{X}({{\mathbb {R}}}^n),L^\infty ({{\mathbb {R}}}^n))_{\theta ,\infty }$$ ( H X ( R n ) , L ∞ ( R n ) ) θ , ∞ , between the Hardy space associated with X , $$H_{X}({{\mathbb {R}}}^n)$$ H X ( R n ) , and the Lebesgue space $$L^\infty ({\mathbb R}^n)$$ L ∞ ( R n ) , is $$WH_{X^{{1}/{(1-\theta )}}}({{\mathbb {R}}}^n)$$ W H X 1 / ( 1 - θ ) ( R n ) , where $$\theta \in (0, 1)$$ θ ∈ ( 0 , 1 ) . All these results are of wide applications. Particularly, when $$X:=M_q^p({{\mathbb {R}}}^n)$$ X : = M q p ( R n ) (the Morrey space), $$X:=L^{\vec {p}}({{\mathbb {R}}}^n)$$ X : = L p → ( R n ) (the mixed-norm Lebesgue space) and $$X:=(E_\Phi ^q)_t({{\mathbb {R}}}^n)$$ X : = ( E Φ q ) t ( R n ) (the Orlicz-slice space), all these results are even new; when $$X:=L_\omega ^\Phi ({\mathbb R}^n)$$ X : = L ω Φ ( R n ) (the weighted Orlicz space), the result on the real interpolation is new and, when $$X:=L^{p(\cdot )}({{\mathbb {R}}}^n)$$ X : = L p ( · ) ( R n ) (the variable Lebesgue space) and $$X:=L_\omega ^\Phi ({{\mathbb {R}}}^n)$$ X : = L ω Φ ( R n ) , the Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ W H X ( R n ) obtained in this article improves the existing results via weakening the assumptions on the Littlewood–Paley functions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b46f2a6d9668b7ab50f5332fa56044e1369bfa60

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In Applied Mathematics

  1. Dachun Yang
  2. Wen Yuan
  3. Yangyang Zhang
  4. Songbai Wang
  5. Xianjie Yan
  6. Sibei Yang
  7. Fan Wang
  8. Der-Chen Chang

In Mathematical Physics

  1. Dachun Yang
  2. Wen Yuan
  3. Yangyang Zhang
  4. Songbai Wang
  5. Xianjie Yan
  6. Sibei Yang
  7. Fan Wang
  8. Der-Chen Chang

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