228 research outputs found
Caffarelli-Kohn-Nirenberg inequalities on Besov and Triebel-Lizorkin-type spaces
We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type
Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey
spaces. More Precisely, we investigate the inequalities \begin{equation*}
\big\|f\big\|_{\dot{k}_{v,\sigma }^{\alpha _{1},r}}\leq
c\big\|f\big\|_{\dot{K}_{u}^{\alpha _{2},\delta }}^{1-\theta
}\big\|f\big\|_{\dot{K}_{p}^{\alpha _{3},\delta _{1}}A_{\beta }^{s}}^{\theta },
\end{equation*} and
\begin{equation*} \big\|f\big\|_{\mathcal{E}_{p,2,u}^{\sigma }}\leq
c\big\|f\big\|_{\mathcal{M}_{\mu }^{\delta }}^{1-\theta
}\big\|f\big\|_{\mathcal{N}_{q,\beta ,v}^{s}}^{\theta }, \end{equation*} with
some appropriate assumptions on the parameters, where is the Herz-type Bessel potential spaces, which are just the
Sobolev spaces if and , and are
Besov or Triebel-Lizorkin spaces if and. To do
these, we study when distributions belonging to these spaces can be interpreted
as functions in . The usual Littlewood-Paley technique,
Sobolev and Franke embeddings, and interpolation theory are the main tools of
this paper. Some remarks on Hardy-Sobolev inequalities are given.Comment: We add Subsection 3.4, Propositions 1-3, Theorem 9 and Appendi
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