A framework of Besov–Triebel–Lizorkin-type spaces via ball quasi-Banach function sequence spaces II: Applications to specific function spaces

In this paper, we apply the framework of Besov–Triebel–Lizorkin-type spaces in terms of ball quasi-Banach function sequence spaces in [J. Sun, D. Yang and W. Yuan, A framework of Besov–Triebel–Lizorkin type spaces via ball quasi-Banach function sequence spaces I: Real-variable characterizations, Math. Ann. 390 (2024) 4283–4360] to several specific function spaces by proving the boundedness of discrete Peetre-type operators and discrete shifts on corresponding sequence spaces and estimating the related critical Peetre decay indices and the related critical (lower or upper) smoothness indices. As a result, we obtain various real-variable characterizations and applications of Besov–Triebel–Lizorkin-type spaces, such as Besov–Triebel–Lizorkin spaces based on ball quasi-Banach function spaces (including Lebesgue spaces, Lorentz spaces, Morrey spaces, Morrey–Lorentz spaces, Besov–Bourgain–Morrey spaces, Triebel–Lizorkin–Bourgain–Morrey spaces, Lorentz–Herz spaces, and generalized Herz spaces), Besov–Triebel–Lizorkin spaces of variable smoothness and integrability, Bourgain–Morrey–Besov–Triebel–Lizorkin spaces, and (generalized) Besov–Triebel–Lizorkin-type spaces. All these obtained results are optimal and several of which improve the existing results, which sufficiently reveal the availability and the power of the aforementioned framework.