Several analytic inequalities in some \(Q\)-spaces (Q653403)

Introduction and statement of main results. In this paper several analytic inequalities in some \(Q\) spaces are studied. First John-Nirenberg type inequalities in \(Q^\beta_\alpha(\mathbb{R}^n)\) \((n\geq 2)\) are established. Then Gagliardo-Nirenberg, Trudinger-Moser and Brezis-Gallouet-Wainger type inequalities in \(Q_\alpha(\mathbb{R}^n)\) are obtained. Here \(Q^\beta_\alpha(\mathbb{R}^n)\) is the set of all measurable complex-valued functions \(f\) on \(\mathbb{R}^n\) satisfying \[ \| f\|_{Q^\beta_\alpha(\mathbb{R}^n)}=\sup_I \Biggl((l(I)^{2(\alpha+ \beta-1)-n} \int_I \int_I {|f(x)- f(y)|^2\over |x-y|^{n+2(\alpha-\beta+ 1)}}\,dx\,dy\Biggr)^{{1\over 2}}<\infty\tag{1.1} \] for \(\alpha\in [-\infty,\beta)\) and \(\beta\in [{1\over 2},1]\), where the supremum is taken over all cubes \(I\) with edge length \(l(I)\) and the edges parallel to the coordinate axes in \(\mathbb{R}^n\). JN type inequalities are classical in modern analysis and widely applied in the theory of partial differential equations. In [Commun. Pure Appl. Math. 14, 415--426 (1961; Zbl 0102.04302)], \textit{F. John} and \textit{L. Nirenberg} proved the JN inequality for \(\text{BMO}(\mathbb{R}^n)\). In this paper, JN type inequalities in \(Q^\beta_\alpha(\mathbb{R}^n)\) are established what implies as a special case Gagliardo-Nirenberg (GN) type inequalities meaning the continuous embeddings such that \(L^r(\mathbb{R}^n)\cap Q_\alpha(\mathbb{R}^n)\subseteq L^p(\mathbb{R}^n)\) for \(-\infty<\alpha<1\) and \(1\leq r\leq p<\infty\). Moreover, from GN type inequalities in \(Q_2(\mathbb{R}^n)\), Trudinger-Moser and Rezis-Gallouet-Wainger type inequalities are deduced. The authors' main results: For any cube \(I\) and integrable function \(f\) on \(I\), we define \[ f(I)= {1\over|I|} \int_I f(x)\,dx,\tag{1.2} \] the mean of \(f\) on \(I\), for \(1\leq q<\infty\), \[ \phi^q_f(I)= {1\over|I|} \int_I |f(x)- f(I)|^q\,dx,\tag{1.3} \] the \(q\)-mean oscillation of \(f\) on \(I\). Let \(D_0= D_0(\mathbb{R}^n)\) be the set of unit cubes whose vertices have integer coordinates, and let, for any integer \(k\in\mathbb{Z}\), \(D_k= D_k(\mathbb{R}^n)= \{2^{-k}I: I\in D_0\}\), then the cubes in \(D= \bigcup^\infty_{-\infty} D_k\) are called dyadic. Furthermore, if \(I\) is any cube, let \(D_k(I)\), \(k\geq 0\), denote the set of the \(2^{kn}\) subcubes of edge length \(2^{-k}l(I)\) obtained by \(k\) succesive bipartitions of each edge of \(I\). Moreover, put \(D(I)= \bigcup^\infty_0 D_k(I)\). For any cube \(I\) and measurable function \(f\) on \(I\), we define \[ \begin{multlined} \psi_{f,\alpha,\beta}(I)= (\ell(I))^{4\beta-4} \sum^\infty_{k=0} \sum_{J\in D_k(I)} 2^{(2(\alpha-\beta+1)-n)k}\phi^2_f(J)=\\ (l(I))^{4\beta-4} \sum_{J\in D(I)}\Biggl({l(J)\over l(I)}\Biggr)^{n-2(\alpha-\beta+k)}\phi^2_f(J).\end{multlined}\tag{1.8} \] We can prove the following proposition by a similar argument applied by \textit{M. Essen, S. Janson, L. Peng} and \textit{J. Xiao} for the case \(\beta= 1\) in [Indiana Univ. Math. J. 49, 575--615 (2000; Zbl 0984.46020)]. Proposition 1.1. Let \(-\infty< \alpha<\beta\) and \(\beta\in({1\over 2}, 1]\). Then \(Q_\alpha(\mathbb{R}^n)\) equals the space of all measurable functions \(f\) on \(\mathbb{R}^n\) such that \(\sup\psi_{f,\alpha,\beta}(I)\) is finite, where \(I\) ranges over all cubes in \(\mathbb{R}^n\). Moreover, the square root of this supremum is a norm on \(Q^\beta_\alpha(\mathbb{R}^n)\), equivalent to \(\| f\|_{Q^\beta_\alpha(\mathbb{R}^n)}\) as defined above. Using this equivalent characterization of \(Q^\beta_\alpha(\mathbb{R}^n)\), we can establish the following JN type inequalities. Theorem 1.2. Let \(-\infty< \alpha<\beta\), \(\beta\in ({1\over 2}, 1]\) and \(0\leq p< 2\). If there exist positive constants \(B\), \(C\) and \(c\), such that, for all cubes \(I\subset\mathbb{R}^n\), and any \(t> 0\), \[ (l(I))^{4\beta-4} \sum^\infty_{k=0} 2^{(2(\alpha-\beta+ 1)- n)k} \sum_{J\in D_k(I)} {m_J(t)\over|J|}\leq B\max\Biggl\{1,\Biggl({C\over t}\Biggr)^p\Biggr\}\exp^{(-ct)},\tag{1.9} \] then \(f\) is a function in \(Q^\beta_\alpha(\mathbb{R}^n)\). Here \(m_I(t)\) is the distribution function of \(f- f(I)\) on the cube \(I\): \[ m_I(t)= |\{x\in I:|f(x)- f(I)|> t\}.\tag{1.10} \] Theorem 1.3. Let \(-\infty< \alpha< \beta\), \(\beta\in({1\over 2}, 1]\) and \(f\in Q^\beta_\alpha(\mathbb{R}^n)\). Then there exist positive constants \(B\) and \(b\), such that \[ \begin{multlined} (l(I))^{4\beta-4} \sum^\infty_{k=0} 2^{(2(\alpha-\beta+ 1)-n)k} \sum_{J\in D_k(I)} {m_J(t))\over|J|}\leq B\max\Biggl\{1,\Biggl({\| f\|_{Q^\beta_\alpha}\over t}\Biggr)^2\Biggr\} \exp\Biggl({-bt\over\| f\|_{Q^\beta_\alpha}}\Biggr)\end{multlined}\tag{1.11} \] holds for \(t\leq\| f\|_{Q^\beta_\alpha(\mathbb{R}^n)}\) and any cubes \(I\subset\mathbb{R}^n\), or for \(t>\| f\|_{Q^\beta_\alpha(\mathbb{R}^n)}\) and cubes \(I\subset \mathbb{R}^n\) with \((l(I))^{2\beta-2}\geq 1\). Moreover, there holds \[ (l(I))^{4\beta- 4}\sum^\infty_{n=0} 2^{(2(\alpha-\beta+1)- n)k}\sum_{J\in D_k(I)} {m_J(t)\over|J|}\leq B\tag{1.12} \] for \(t>\| f\|_{Q^\beta_\alpha(\mathbb{R}^n)}\) and cubes \(I\subset\mathbb{R}^n\) with \((l(I))^{2\beta-2}< 1\). If \(\alpha\in (-\infty,\beta-1)\), then all \(Q^\beta_\alpha(\mathbb{R}^n)\) equal to \(Q^\beta_{-{n\over 2}+\beta-1}(\mathbb{R}^n):= \text{BMO}^\beta(\mathbb{R}^n)\). Thus, when \(k=0\) and \(\alpha=-{n\over 2}+\beta-1\), (1.11) implies a special JN type inequality that is, for \(f\in L^2(\mathbb{R}^n)\cap \text{BMO}^\beta(\mathbb{R}^n)\) and \(t\leq\| f\|_{\text{BMO}^\beta(\mathbb{R}^n)}\), \[ |\{x\in\mathbb{R}^n: |f|> t\}|\leq {B\| f\|^2_{L^2(\mathbb{R}^n)}\over t^2} \exp\Biggl({-bt\over\| f\|_{\text{BMO}^\beta(\mathbb{R}^n)}}\Biggr).\tag{1.13} \] When \(t>\| f\|_{\text{BMO}^\beta(\mathbb{R}^n)}\), we get a weaker form (1.13). Proposition 1.4. Let \(\beta\in ({1\over 2},1]\). If \(f\in \text{BMO}^\beta(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)\), then (i) (1.13) holds for all \(t\leq\| f\|_{\text{BMO}^\beta(\mathbb{R}^n)}\); (ii) \(|\{x\in\mathbb{R}^n: f(x)> t\}|\leq {B\| f\|^2_{L^2(\mathbb{R}^2)}\over\| f\|^2_{\text{BMO}^\beta(\mathbb{R}^n)}}\) holds for all \(t>\| f\|_{\text{BMO}^\beta(\mathbb{R}^n)}\). When \(\beta= 1\) and \(t>\| f\|_{\text{BMO}(\mathbb{R}^n)}\), (1.13) also holds and implies the following GN type inequalities in \(Q_\alpha(\mathbb{R}^n)\) which can also be deduced from [J. Math. Anal. Appl. 303, No. 2, 696-.698 (2005; Zbl 1095.42012 ), Theorem 2] and [Indiana Univ. Math. J. 49, 575--615 (2000; Zbl 0984.46020), Theorem 2.3]: for \(-\infty< \alpha< 1\) and \(1\leq r\leq p<\infty\), \[ \| f\|_{L^p(\mathbb{R}^n)}\leq C_n p\| f\|^{r/p}_{L^r(\mathbb{R}^n)}\| f\|^{1-r/p}_{Q_\alpha(\mathbb{R}^n)},\tag{1.15} \] for \(f\in L^r(\mathbb{R}^n)\cap Q_\alpha(\mathbb{R}^n)\). As an application of (1.15), we establish the Trudinger-Moser type inequality which implies a generalized JN type inequality. Theorem 1.5. (i) There exists a positive constant \(\gamma_n\) such that for every \(0<\xi<\gamma_n\) \[ \int_{\mathbb{R}^n} \psi_p\Biggl(\xi\Biggl({|f(x)|\over\| f\|_{Q_\alpha(\mathbb{R}^n)}}\Biggr)\Biggr)\,dx\leq C_{n,\xi}\Biggl({\| f\|_{L^p(\mathbb{R}^n)}\over\| f\|_{Q_\alpha(\mathbb{R}^n)}}\Biggr)^p\tag{1.16} \] holds for all \(f\in L^p(\mathbb{R}^n)\cap Q_\alpha(\mathbb{R}^n)\) with \(1< p<\infty\) and \(-\infty< \alpha< 1\). Here \(\phi_p\) is the function defined by \[ \phi_p(t)= e^t- \sum_{j< p,j\in \mathbb N\cup\{0\}} {t^j\over j!},\quad t\in\mathbb{R}. \] (ii) There exists a positive constant \(\gamma_n\) such that \[ |\{x\in\mathbb{R}^n: |f|> t\}|\leq C_n{\| f\|^2_{L^2(\mathbb{R}^n)}\over\| f\|^2_{Q_\alpha(\mathbb{R}^n)}}\cdot {1\over(\exp({t\gamma_n\over\| f\|_{Q_\alpha(\mathbb{R}^n)}})- 1-{t\gamma_n\over\| f\|_{Q_\alpha(\mathbb{R}^n)}})}\tag{1.17} \] holds for all \(t> 0\) and \(f\in L^2(\mathbb{R}^n)\cap Q_\alpha(\mathbb{R}^n)\) with \(-\infty< \alpha< 1\). In particular, we have \[ |\{x\in \mathbb{R}^n: |f|> t\}|\leq C_n {\| f\|^2_{L^2(\mathbb{R}^n)}\over\| f\|^2_{Q_\alpha(\mathbb{R}^n)}} \exp\Biggl(-{t\gamma_n\over\| f\|_{Q_\alpha(\mathbb{R}^n)}}\Biggr) \] holds for all \(t>\| f\|_{Q_\alpha(\mathbb{R}^n)}\) and \[ f\in L^2(\mathbb{R}^n)\cap Q_\alpha(\mathbb{R}^n)\quad\text{with }-\infty< \alpha< 1. \] We can also get the following Brezis-Gallouet-Weinger type inequalities. Proposition 1.6. For every \(1< q<\infty\) and \(n/q< s<\infty\), we have \[ \| f\|_{L^\infty(\mathbb{R}^n)}\leq C_{n,p,q,s}(1+(\| f\|_{L^p(\mathbb{R}^n)}+\| f\|_{Q_\alpha(\mathbb{R}^n)}\log(e+\|(-\Delta)^{s/2} f\|_{L^q(\mathbb{R}^n)})) \] holds for all \((-\Delta)^{s/2}f\in L^q(\mathbb{R}^n)\) satisfying \(f\in L^p(\mathbb{R}^n)\cap Q_\alpha(\mathbb{R}^n)\) when \(1\leq p<\infty\) and \(-\infty<\alpha<1\).