Let $X$ be a ball Banach function space on ${mathbb R}^n$. In this article, under the mild assumption that the Hardy--Littlewood maximal operator is bounded on the associated space $X$ of $X$, the authors prove that, for any $fin C_{mathrm{c}}^2({mat
hbb R}^n)$, $$sup_{lambdain(0,infty)}lambdaleft |left|left{yin{mathbb R}^n: |f(cdot)-f(y)| >lambda|cdot-y|^{frac{n}{q}+1}right}right|^{frac{1}{q}} right|_Xsim | abla f|_X$$ with the positive equivalence constants independent of $f$, where $qin(0,infty)$ is an index depending on the space $X$, and $|E|$ denotes the Lebesgue measure of a measurable set $Esubset {mathbb R}^n$. Particularly, when $X:=L^p({mathbb R}^n)$ with $pin [1,infty)$, the above estimate holds true for any given $qin [1, p]$, which when $q=p$ is exactly the recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which even when $q< p$ is new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and 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 Poincare inequality, the extrapolation, the exact operator norm on $X$ of the Hardy--Littlewood maximal operator, and the geometry of $mathbb{R}^n.$
In this article, the authors give a survey on the recent developments of both the John--Nirenberg space $JN_p$ and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, $VJN_p$, and $CJN_p$ on $mathbb{R}^n$ or a given cube $Q_0sub
setmathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.
Some Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $tau$, $p$, $q$ and $n$ of
this characterization. Therefore, the authors measure the regularity of the characteristic function $mathcal X$ of the unit cube in $mathbb{R}^n$ via the Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $langle f, mathcal{X} rangle$ generates a continuous linear functional on $B^{s,tau}_{p,q}(mathbb{R}^n)$.
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ a one to one operator of type $omega$ in $L^2({mathbb R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic fu
nctional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W!H_L^{p(cdot)}(mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good an
d bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $mathbb R^n$, $W!H^{p(cdot)}(mathbb R^n)$, via
the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W!H^{p(cdot)}(mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_{lambda}^ast$-function. As an application, the authors establish the boundedness of convolutional $delta$-type and non-convolutional $gamma$-order Calderon-Zygmund operators from $H^{p(cdot)}(mathbb R^n)$ to $W!H^{p(cdot)}(mathbb R^n)$ including the critical case $p_-={n}/{(n+delta)}$, where $p_-:=mathopmathrm{ess,inf}_{xin rn}p(x).$
Let $Dinmathbb{N}$, $qin[2,infty)$ and $(mathbb{R}^D,|cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, via an auxiliary function space $mathrm{WE}^{1,,q}(mathbb R^D)$ defined via wavelet expansion
s, the authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$. As a consequence, the authors obtain the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$. Finally, the authors give an explicit example to show that $dot{F}^0_{1,,q}(mathbb{R}^D)$ is strictly contained in $mathrm{WE}^{1,,q}(mathbb{R}^D)$ and, by duality, $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ is strictly contained in $dot{F}^0_{infty,,q}(mathbb{R}^D)$. Although all results when $D=1$ were obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the case $Dge2$, which needs some new skills.
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally $log$-Holder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estima
tes. Let $H_L^{p(cdot)}(mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels ${e^{-t^2L}}_{tin (0,infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(cdot)}(mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(cdot)}(mathbb R^n)$ under an additional assumption that the heat kernels of $L$ have the Holder regularity.
Let $L$ be a one-to-one operator of type $omega$ in $L^2(mathbb{R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(cdot): mathbb{R}^nto(0,,1]$ be a variable exponent
function satisfying the globally log-H{o}lder continuous condition. In this article, the authors introduce the variable Hardy space $H^{p(cdot)}_L(mathbb{R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{p(cdot)}_L(mathbb{R}^n)$. Then the authors show that the dual space of $H^{p(cdot)}_L(mathbb{R}^n)$ is the BMO-type space ${rm BMO}_{p(cdot),,L^ast}(mathbb{R}^n)$, where $L^ast$ denotes the adjoint operator of $L$. In particular, when $L$ is the second order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H^{p(cdot)}_L(mathbb{R}^n)$ and show that the fractional integral $L^{-alpha}$ for $alphain(0,,frac12]$ is bounded from $H_L^{p(cdot)}(mathbb{R}^n)$ to $H_L^{q(cdot)}(mathbb{R}^n)$ with $frac1{p(cdot)}-frac1{q(cdot)}=frac{2alpha}{n}$ and the Riesz transform $ abla L^{-1/2}$ is bounded from $H^{p(cdot)}_L(mathbb{R}^n)$ to the variable Hardy space $H^{p(cdot)}(mathbb{R}^n)$.
Let $L$ be a linear operator on $L^2(mathbb R^n)$ generating an analytic semigroup ${e^{-tL}}_{tge0}$ with kernels having pointwise upper bounds and $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder conti
nuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(cdot)}(mathbb R^n)$, and the BMO-type space ${mathrm{BMO}}_{p(cdot),L}(mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(cdot)}(mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(cdot)}(mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$.
