Hello esteemed members,
I am currently grappling with some properties of non-homogeneous Besov spaces and am seeking guidance or references to aid my understanding.
Embedding of Besov Spaces: I aim to establish the embedding $B^{\alpha}_{p,q_2} \subset B^{\alpha'}_{p,q_1}$ given the conditions $ \alpha' < \alpha $ and $ q_1 \leq q_2 $. I am curious if this can be achieved without resorting to interpolation theory. If not, any recommendations or references regarding the theory of interpolation, especially in the context of non-homogeneous Besov spaces, would be greatly appreciated.
Besov Space and Hölder Continuity: Additionally, I am working to show that the non-homogeneous Besov space $B^{\alpha}_{\infty,\infty}$ for $ \alpha \in (0,1) $ coincides with the space of bounded $ \alpha $-Hölder continuous functions. My primary reference suggests employing the Bernstein-type inequalities for this endeavor. I would be grateful for any insights or alternative approaches to this proof.
Your expertise and suggestions will be invaluable to my research. Thank you in advance for taking the time to assist me.