Relations among Besov-type spaces, Triebel–Lizorkin-type spaces and generalized Carleson measure spaces

By Dachun Yang, Wen Yuan. 2011

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In this article, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel–Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 919–926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel–Lizorkin-type space and the Besov-type space coincide with certain Triebel–Lizorkin space, which answers a question posed in Remark 6.11(i) of Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/13ab3c62472896ff31b26096e043b8f72efa8698

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Citations in the corpus

Triebel-Lizorkin-Type Spaces with Variable Exponents (2015, background): “For more properties of these spaces, we refer the reader to [63, 64, 78, 79].”
Generalized Morrey spaces and trace operator (2016, background): “5]. Yang, Yuan and Zhuo investigated the boundedness property of the Fourier multiplier precisely in [94]. Yang and Yuan characterized As,ρ pq(Rn) in terms of the Peetre maximal operator in [87]. See [89, 96] for more recent advances in Triebel-Lizorkin type spaces, and see [17, 51] for the extensio”
Besov-Type Spaces with Variable Smoothness and Integrability (2015, background): “Various properties and equivalent characterizations of Besov-type and Triebel-Lizorkin-type spaces, including smoothness atomic, molecular or wavelet decompositions, characterizations, respectively, via differences, oscillations, Peetre maximal functions, Lusin area functions or g∗ λ functions, have”
New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets (2011, unknown): “Moreover, recently, it was proved in [60] that, for all s ∈ R, both the spaces Ḃ p,q (Rn) and Ḟ s,τ p,q (Rn) coincide with Ḃ s+n(τ−1/p) ∞,∞ (Rn) if τ ∈ (1/p,∞) and q ∈ (0,∞), or τ ∈ [1/p,∞) and q = ∞.”