Smoothness spaces related to Morrey spaces -- a survey. II (Q2857705)

For Part I, see [ibid. 3, No. 3, 110--149 (2012; Zbl 1274.46074)].NEWLINENEWLINEThis paper is the second part of the long and interesting survey on smoothness spaces related to Morrey spaces \(\mathcal{M}^p_u\), which contain eight scales of function spaces: the Nikol'skii-Besov type space \(B_{p,q}^{s,\tau}\), the Lizorkin-Triebel type space \(F_{p,q}^{s,\tau}\), the Nikol'skii-Besov-Morrey space \(\mathcal{N}_{p,q,u}^s\), the Lizorkin-Triebel-Morrey space \(\mathcal{E}_{p,q,u}^s\), as well as their local versions: \(B^{s,\tau}_{p,q,\text{unif}}\), \(F^{s,\tau}_{p,q,\text{unif}}\), \({N}_{p,q,u}^s\) and \({E}_{p,q,u}^s\). Some interpolation properties of these spaces (with fixed \(p\)), the related Gagliardo-Nirenberg type inequalities and some embedding properties of these spaces are presented.NEWLINENEWLINEThe first main topic is the (real) interpolation of Nikol'skii-Besov type spaces and Lizorkin-Triebel type spaces for fixed \(p\) and fixed \(\tau\) in Section 2. Precisely, let \(\theta\in(0,1)\), \(s_0,s_1\in\mathbb R\) with \(s_0<s_1\), \(q,q_0,q_1\in(0,\infty]\), \(s=(1-\theta)s_0+\theta s_1\) and \(\frac1q=\frac{1-\theta}{q_0}+\frac{\theta}{q_1}\). Via studying the approximation spaces of Nikol'skii-Besov type spaces, the author proves that, if \(p\in(0,\infty)\) and \(\tau\in[0,1/p]\), then \(\mathcal{B}_{p,q}^{s,\tau}=(B_{p,q_0}^{s_0,\tau},B_{p,q_1}^{s_1,\tau})_{\theta,q}\); if \(0<u\leq p\leq\infty\), then \(\mathcal{N}_{p,q,u}^s=(A_{p,q_0,u}^{s_0},\mathcal{A}_{p,q_1,u}^{s_1})_{\theta,q}\), where \(\mathcal{B}_{p,q}^{s,\tau}\) is a variant of \({B}_{p,q}^{s,\tau}\) defined via replacing the summation \(\sum_{j=\max\{j_Q,0\}}^\infty\) in the definition of \({B}_{p,q}^{s,\tau}\) by \(\sum_{j=0}^\infty\) and by interchanging supremum taking and summation, and \(A,\mathcal{A}\in\{\mathcal{E},\mathcal{N}\}\) (\(p\in(0,\infty)\) if either \(A=\mathcal{E}\) or \(\mathcal{A}=\mathcal{E}\)). When the parameters \(p,u,s\) are fixed, it is proved that, for \(1\leq u\leq p\leq\infty\) and \(s\in\mathbb{R}\), \(\mathcal{N}_{p,q,u}^s=(\mathcal{N}_{p,q_0,u}^s,\mathcal{N}_{p,q_1,u}^s)_{\theta,q}\). When the parameters \(s,q,u\) are changeable, only one direction is known: for \(0<u_0\leq u_1\leq p\leq\infty\) and \(\frac 1u=\frac{1-\theta}{u_0}+\frac{\theta}{u_1}\), it holds that \((\mathcal{M}_{u_0}^p,\mathcal{M}_{u_1}^p)_{\theta,u}\hookrightarrow \mathcal{M}_{u}^p\); if \(1\leq u\leq p\leq\infty\), \(\frac 1u=\frac{ 1-\theta}{u_0}+\frac {\theta}{u_1}\) and \(u=q\), then \((\mathcal{N}_{p,q_0,u_0}^{s_0},\mathcal{N}_{p,q_1,u_1}^{s_1})_{\theta,q} \hookrightarrow \mathcal{N}_{p,q,u}^s\).NEWLINENEWLINEThe Gagliardo-Nirenberg type inequalities on Lizorkin-Triebel type spaces and Besov type spaces are established in Section 3. Let \(q,q_0,q_1\in(0,\infty]\), \(-\infty<s_0<s_1<\infty\), \(\tau_0,\tau_1\in[0,\infty)\), \(\tau=(1-\theta)\tau_0+\theta\tau_1\), \(s=(1-\theta)s_0+\theta s_1\) and \(\theta\in(0,1)\). It is proved that, if \(p_0,p_1\in(0,\infty)\), \(\frac 1p=\frac {1-\theta}{p_0}+\frac{\theta}{p_1}\), then there exists a positive constant \(c\) such that, for all tempered distributions \(f\), NEWLINE\[NEWLINE \|f|F_{p,q}^{s,\tau}\|\leq c\|f|F_{p_0,q_0}^{s_0,\tau_0}\|^{1-\theta} \|f|F_{p_1,q_1}^{s_1,\tau_1}\|^\theta;NEWLINE\]NEWLINE if \(p_0\in(0,\infty)\) and \(\frac 1p=\frac{1-\theta}{p_0}\), then there exists a positive constant \(c\) such that for all tempered distributions \(f\), NEWLINE\[NEWLINE \|f|F_{p,q}^{s,\tau}\|\leq c\|f|F_{p_0,q_0}^{s_0,\tau_0}\|^{1-\theta} \|f|B_{\infty,\infty}^{s_1,\tau_1}\|^\theta;NEWLINE\]NEWLINE if \(p_0,p_1\in(0,\infty]\) and \(\frac 1q=\frac{1-\theta}{q_0}+\frac{\theta}{q_1}\), then there exists a positive constant \(c\) such that for all tempered distributions \(f\), NEWLINE\[NEWLINE \|f|B_{p,q}^{s,\tau}\|\leq c\|f|B_{p_0,q_0}^{s_0,\tau_0}\|^{1-\theta} \|f|B_{p_1,q_1}^{s_1,\tau_1}\|^\theta.NEWLINE\]NEWLINENEWLINENEWLINESection 4 of this paper mainly concerns embeddings. It is proved that, if \(s_0\in\mathbb R\), \(0<p_0<p_1<\infty\), \(q_0,q_1\in(0,\infty]\), \(\tau_0\in[0,\infty)\), \(\tau=\frac{p_0\tau_0}{p_1}\) and \(s=s_0+d(\tau_0-\frac1{p_0})-d(\tau-\frac1{p_1})\), then \(F_{p_0,q_0}^{s_0,\tau_0}\hookrightarrow F_{p_1,q_1}^{s,\tau}\). A similar result on Besov type spaces is also obtained. The embeddings of Besov type and Lizorkin-Triebel type spaces into \(C_{ub}\), i.\,e., the collection of all complex-valued, uniformly continuous and bounded functions on \(\mathbb R^d\), are also considered.NEWLINENEWLINEIn Section 5, the author recalls three further approaches to smoothness spaces related Morrey spaces including Triebel's local spaces, the approach of \textit{L. I. Hedberg} and \textit{Y. Netrusov} [``An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation'', Mem. Am. Math. Soc. 882, 97 p. (2007; Zbl 1186.46028)], and a recent general framework of Nikol'skii-Bosev-Lizorkin-Triebel spaces which does not require the boundedness of maximal operators [\textit{Y.-Y. Liang} et al., ``A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces'', Diss. Math. 489, 114 p. (2013; Zbl 1283.46027)]. Finally, in Section~6, a series of open problems concerning further properties and generalizations of the scales \(B_{p,q}^{s,\tau}\) and \(F_{p,q}^{s,\tau}\) are presented.