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In this article, we establish a characterization of the set $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ of all pointwise multipliers of Besov spaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with only logarithmic smoothness $b\in\mathbb{R}$ in the special cases $p=1$ and $p=\infty$. As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions $e^{ik\cdot x}$ with $k\in\mathbb{Z}^n$ and $x\in\mathbb{R}^n$, are pointwise multipliers of $B^{0,b}_{1,\infty}(\mathbb{R}^n)$ and $B^{0,b}_{\infty,\infty}(\mathbb{R}^n)$, respectively; furthermore, we obtain the explicit estimates of $\|e^{ik \cdot x}\|_{M(B^{0,b}_{1,\infty}(\mathbb{R}^n))}$ and $\|e^{ik \cdot x}\|_{M(B^{0,b}_{\infty,\infty}(\mathbb{R}^n))}$. In the case that $p\in(1,\infty)$, we give some sufficient conditions and some necessary conditions of the pointwise multipliers of $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ and a complete characterization of $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ is still open. However, via a different method, we are still able to accurately calculate $\|e^{ik \cdot x}\|_{M(B^{0,b}_{p,\infty}(\mathbb{R}^n))}$, $k\in\mathbb{Z}^n$, in this situation. The novelty of this article is that most of the proofs are constructive and these constructions strongly depend on the logarithmic structure of Besov spaces under consideration., Comment: 64 pages; Submitted

Let $X$ be a ball quasi-Banach function space satisfying some mild assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In this article, the authors introduce both the Hardy space $H_X(\mathbb{R}^{n+1}_+)$ of harmonic functions and the Hardy space $\mathbb{H}_X(\mathbb{R}^{n+1}_+)$ of harmonic vectors, associated with $X$, and then establish the isomorphisms among $H_X(\mathbb{R}^n)$, $H_{X,2}(\mathbb{R}^{n+1}_+)$, and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$, where $H_{X,2}(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$ are, respectively, certain subspaces of $H_X(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_X(\mathbb{R}^{n+1}_+)$. Using these isomorphisms, the authors establish the first order Riesz transform characterization of $H_X(\mathbb{R}^n)$. The higher order Riesz transform characterization of $H_X(\mathbb{R}^n)$ is also obtained. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the variable Hardy space, the mixed-norm Hardy space, the local generalized Herz-Hardy space, and the mixed-norm Herz-Hardy space., Comment: 43 pages, Submitted

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range of generality and can be applied to the weighted Hardy space, the variable Hardy space, the mixed-norm Hardy space, the Orlicz--Hardy space, the Orlicz-slice Hardy space, and the Morrey--Hardy space, associated with $L$. In particular, even when $L$ is a second order divergence form elliptic operator, both the maximal function and the Riesz transform characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey--Hardy space, associated with $L$, obtained in this article, are totally new., Comment: 52 pages; Submitted

The stability and equation of state of quark matter are studied within both two-flavor and (2+1)-flavor Nambu-Jona-Lasinio (NJL) models including the vector interactions. With a free parameter $\alpha$, the Lagrangian is constructed by two parts, the original NJL Lagrangian and the Fierz transformation of it, as $L=(1-\alpha) L_{\rm{NJL}}+\alpha L_{\rm{Fierz}}$. We find that there is a possibility for both $ud$ nonstrange and $uds$ strange matter being absolute stable, depending on the interplay of the confinement with quark vector interaction and the exchange interaction channels. The calculated quark star properties can reconcile with the recently measured masses and radii of PSR J0030+0451 and PSR J0740+6620, as well as the tidal deformability of GW170817. Furthermore, the more strongly-interacting quark matter in the nonstrange stars allows a stiffer equation of state and consequently a higher maximum mass ($\sim2.7\, M_{\odot}$) than the strange ones ($\sim2.1\, M_{\odot}$). The sound velocities in strange and nonstrange quark star matter are briefly discussed compared to those of neutron star matter., Comment: 13 pages, 9 figures, 2 tables, Phys. Rev. D (2022) accepted

In this article, for an optimal range of the smoothness parameter $s$ that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Haj{\l}asz--Triebel--Lizorkin spaces $M^s_{p,q}$ and Haj{\l}asz--Besov spaces $N^s_{p,q}$ in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results in this article, the authors obtain a new characterization for Sobolev embedding and extension domains in the context of general doubling metric measure spaces., Comment: arXiv admin note: text overlap with arXiv:2202.06389

In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter $s$. An interesting facet of this work is how the range of $s$ for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-Besov spaces in the context of quasi-metric spaces, the main results in this article improve known work even for Sobolev spaces in the metric setting.

In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Haj\l asz-Sobolev spaces, Haj\l asz-Besov and Haj\l asz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of the inhomogeneous version is new even when the underlying space is an RD-space., Comment: arXiv admin note: text overlap with arXiv:2012.13035, arXiv:2103.02390, arXiv:2103.08227

Let $(\mathbb{X},d,\mu)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calder\'{o}n--Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki--Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $\mathbb{X}$, which closely connect with the geometrical properties of $\mathbb{X}$, to overcome the difficulties caused by the absence of both the triangle inequality of $\|\cdot\|_{X(\mathbb{X})}$ and the reverse doubling assumption of the measure $\mu$ under consideration, and also use the relation between the convexification of $X(\mathbb{X})$ and the weak space $WX(\mathbb{X})$ associated with $X(\mathbb{X})$ to prove that the infinite summation of atoms converges in the space of distributions on $\mathbb{X}$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).

Let $n\ge2$ and $\Omega\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $\Omega$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse H\"older inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ symmetric part and small $\mathrm{BMO}$ anti-symmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, $C^1$ domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz--)Morrey spaces, (Musielak--)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix., Comment: 41 pages; Submitted. arXiv admin note: substantial text overlap with arXiv:2003.07609

To shed some light on the John--Nirenberg space, the authors in this article introduce the John--Nirenberg-$Q$ space via congruent cubes, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$, which when $p=\infty$ and $q=2$ coincides with the space $Q_\alpha(\mathbb{R}^n)$ introduced by Ess\'en et al. in [Indiana Univ. Math. J. 49 (2000), 575-615]. Moreover, the authors show that, for some particular indices, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ coincides with the congruent John--Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into $JNQ^\alpha_{p,q}(\mathbb{R}^n)$. Furthermore, the authors characterize $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of `almost increasing' set function introduced in this article, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length., Comment: 38 pages, submitted

Let $(\mathcal{X}, \rho, \mu)$ be a metric measure space of homogeneous type which supports a certain Poincar\'e inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with compact support satisfying that $\operatorname{Lip} f:=\limsup _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ is also a continuous function with compact support and $\operatorname{Lip} f=\lim _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ converges uniformly. Let $p \in[1,\infty)$. In this article, the authors prove that, for any $f\in\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$, \begin{align*} &\sup_{\lambda\in(0,\infty)}\lambda^p\int_{\mathcal{X}} \mu\left(\left\{y\in \mathcal{X}:\ |f(x)-f(y)|>\lambda \rho(x,y) [V(x,y)]^{\frac 1p}\right\}\right)\, d\mu(x)\\ &\quad\sim \int_{{\mathcal{X}}} [\operatorname{Lip}f(x)]^p \,d\mu(x) \end{align*} with the positive equivalence constants independent of $f$, where $V(x,y):=\mu(B(x,\rho(x,y)))$. This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the $n$-dimensional Euclidean space ${\mathbb R}^n$ to $\mathcal{X}$. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo--Nirenberg inequalities in $\mathcal{X}$. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, ${\mathbb R}^n$ with weighted Lebesgue measure and the complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincar\'e inequality., Comment: 25 pages; Submitted. arXiv admin note: text overlap with arXiv:2109.04638

Let $({\mathcal X},\rho,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of $H_{Y}^*({\mathcal X})$, which proves to be a ball Campanato-type function space associated with $Y({\mathcal X})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of $\mu$ and the triangle inequality of $\rho$, the authors cleverly construct admissible sequences of balls, and fully use the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of $H_{Y}^*({\mathcal X})$, the authors further prove that $Y({\mathcal X})$ can be embedded into the weighted Lebesgue space with certain special weight, and then can fully use the known results of the weighted Lebesgue space.

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{\lambda\in(0,\infty)}\lambda\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >\lambda|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\infty)$ is related to $X$ and $|\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >\lambda|\cdot-y|^{\frac{n}{q}+1}\}|$ is the Lebesgue measure of the set under consideration. In particular, when $X:=L^p({\mathbb R}^n)$ with $p\in [1,\infty)$, the above formulae hold true for any given $q\in (0,\infty)$ with $n(\frac{1}{p}-\frac{1}{q})<1$, which when $q=p$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo--Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincar\'e inequality, the extrapolation, the exact operator norm on $X'$ of the Hardy--Littlewood maximal operator, and the exquisite geometry of $\mathbb{R}^n.$, Comment: 68 pages, Submitted

Let $p\in[1,\infty]$, $q\in[1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $\alpha\in\mathbb{R}$. In this article, the authors first find a reasonable version $\widetilde{I}_{\beta}$ of the (generalized) fractional integral $I_{\beta}$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, which coincides with the Campanato space $\mathcal{C}_{\alpha,q,s}(\mathbb{R}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_{\beta}$. Then the authors prove that $\widetilde{I}_{\beta}$ is bounded from $JN_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ to $JN_{(p,q,s)_{\alpha+\beta/n}}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if $I_{\beta}$ has the vanishing moments up to order $s$. The obtained result is new even when $p=\infty$ and $s\in\mathbb{N}$. Moreover, the authors show that $I_{\beta}$ can be extended to a unique continuous linear operator from the Hardy-kind space $HK_{(p,q,s)_{\alpha+\beta/n}}^{\mathrm{con}}(\mathbb{R}^n)$, the predual of $JN_{(p',q',s)_{\alpha+\beta/n}}^{\mathrm{con}}(\mathbb{R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, to $HK_{(p,q,s)_{\alpha}}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if $I_{\beta}$ has the vanishing moments up to order $s$. The proof of the latter boundedness strongly depends on the dual relation $(HK_{(p,q,s)_{\alpha}}^{\mathrm{con}}(\mathbb{R}^n))^* =JN_{(p',q',s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, the properties of molecules of $HK_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, and a crucial criterion for the boundedness of linear operators on $HK_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$., Comment: 38 pages; Front. Math. China. (to appear). arXiv admin note: text overlap with arXiv:2108.01517

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and assume that the Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$, and let $q\in[1,\infty)$ and $d\in(0,\infty)$. In this article, the authors prove that, for any $f\in \mathcal{L}_{X,q,0,d}(\mathbb{R}^n)$ (the ball Campanato-type function space associated with $X$), the Littlewood--Paley $g$-function $g(f)$ is either infinite everywhere or finite almost everywhere and, in the latter case, $g(f)$ is bounded on $\mathcal{L}_{X,q,0,d}(\mathbb{R}^n)$. Similar results for both the Lusin-area function and the Littlewood--Paley $g_\lambda^*$-function are also obtained. All these results have a wide range of applications. Particularly, even when $X$ is the weighted Lebesgue space, or the mixed-norm Lebesgue space, or the variable Lebesgue space, or the Orlicz space, or the Orlicz-slice space, all these results are new. The proofs of all these results strongly depend on several delicate estimates of Littlewood--Paley operators on the mean oscillation of the locally integrable function $f$ on $\mathbb{R}^n$. Moreover, the same ideas are also used to obtain the corresponding results for the special John--Nirenberg--Campanato space via congruent cubes., Comment: 46 pages; Submitted

Let $p\in[1,\infty]$, $q\in(1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $\alpha\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calder\'on--Zygmund operator $T$ on $JN_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, the special John--Nirenberg--Campanato space via congruent cubes, which coincides with the Campanato space $\mathcal{C}_{\alpha,q,s}(\mathbb{R}^n)$ when $p=\infty$. Then the authors prove that $\widetilde T$ is bounded on $JN_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if, for any $\gamma\in\mathbb{Z}_+^n$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$, which is a well-known assumption. To this end, the authors find an equivalent version of this assumption. Moreover, the authors show that $T$ can be extended to a unique continuous linear operator on the Hardy-kind space $HK_{(p,q,s)_{\alpha}}^{\mathrm{con}}(\mathbb{R}^n)$, the predual space of $JN_{(p',q',s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, if and only if, for any $\gamma\in\mathbb{Z}_+^n$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$., Comment: 47 pages; Submitted