tent spaces
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2021 ◽  
Vol 56 (5) ◽  
pp. 295-306
Author(s):  
R. Qian ◽  
S. Li
Keyword(s):  




2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Marco Fraccaroli

AbstractWe study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .





2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Dan Qu ◽  
Xiangling Zhu ◽  
Ruishen Qian

The boundedness and compactness of the inclusion mapping from Besov spaces to tent spaces are studied in this paper. Meanwhile, the boundedness, compactness, and essential norm of the Volterra integral operator T g from Besov spaces to a class of general function spaces are also investigated.



2021 ◽  
Vol 6 (1) ◽  
pp. 698-711
Author(s):  
Ruishen Qian ◽  
◽  
Xiangling Zhu ◽  


2020 ◽  
pp. 1-12
Author(s):  
Ruishen Qian ◽  
Xiangling Zhu

Abstract In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.



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