We study the space BMO in the general setting of a measure space $\mathbb{X}$ with a fixed collection $\mathscr{G}$ of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in $\mathscr{G}$. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space $\mathbb{X}$ for BMO to be a Banach space modulo constants. We also develop the notion of a Denjoy family $\mathscr{G}$, which guarantees that functions in BMO satisfy the John-Nirenberg inequality on the elements of $\mathscr{G}$.
Global and local weighted Gagliardo-Nirenberg inequalities with doubling measures are established. These inequalities are key ingredients for the regularity theory and existence of strong solutions for strongly coupled parabolic and elliptic systems which are degenerate or singular because of the unboundedness of dependent and independent variables.
In this paper we introduce the generalized BMO martingale spaces by stopping time sequences, which enable us to characterize the dual spaces of martingale Hardy-Lorentz spaces $H_{p,q}^s$ for $0<p\leq1, 1<q<\infty$. Moreover, by duality we obtain a John-Nirenberg theorem for the generalized BMO martingale spaces when the stochastic basis is regular. We also extend the boundedness of fractional integrals to martingale Hardy-Lorentz spaces.
This work explores new deep connections between John-Nirenberg type inequalities and Muckenhoupt weight invariance for a large class of $BMO$-type spaces. The results are formulated in a very general framework in which $BMO$ spaces are constructed using a base of sets, used also to define weights with respect to a non-negative measure (not necessarily doubling), and an appropriate oscillation functional. This includes as particular cases many different function spaces on geometric settings of interest. As a consequence the weight invariance of several $BMO$ and Triebel-Lizorkin spaces consider…
We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John-Nirenberg space and provide a method to construct nontrivial functions in the dyadic John-Nirenberg space. Moreover, we prove that the John-Nirenberg space is complete. Several open problems are also discussed.
We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón-Zygmund decomposition and a good-$λ$ inequality for medians. A John-Nirenberg inequality up to the boundary is proven by us…
In this paper, we develop an abstract framework for John-Nirenberg inequalities associated to BMO-type spaces. This work can be seen as the sequel of [5], where the authors introduced a very general framework for atomic and molecular Hardy spaces. Moreover, we show that our assumptions allow us to recover some already known John-Nirenberg inequalities. We give applications to the atomic Hardy spaces too.
A generalization of the theory of Y. Brudnyi \cite{yuri}, and A. and Y. Brudnyi \cite{BB20a}, \cite{BB20b}, is presented. Our construction connects Brudnyi's theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak--Kuznetsov. Our spaces shed light on the structure of the John--Nirenberg spaces. We show that $SJN_{p}$ (sparse John--Nirenberg space) coincides with $L^{p},1<p<\infty.$ This characterization yields the John--Nir…
We develop some techniques for studying various versions of the function space BMO. Special cases of one of our results give alternative proofs of the celebrated John- Nirenberg inequality and of related inequalities due to John and to Wik. Our approach enables us to pose a simply formulated "geometric" question, for which an affirmative answer would lead to a version of the John-Nirenberg inequality with dimension free constants. A more detailed summary of the main ideas and results of this paper can be found at http://www.math.technion.ac.il/~mcwikel/bmo/CwikSaghShvaSummary.pdf
We consider BMO spaces of operator-valued functions, among them the space of operator-valued functions $B$ which define a bounded paraproduct on $L^2(H)$. We obtain several equivalent formulations of $\|π_B\|$ in terms of the norm of the "sweep" function of $B$ or of averages of the norms of martingales transforms of $B$ in related spaces. Furthermore, we investigate a connection between John-Nirenberg type inequalities and Carleson-type inequalities via a product formula for paraproducts and deduce sharp dimensional estimates for John-Nirenberg type inequalities.
We extend to n-dimensions a characterization of the Marcinkiewicz $L(p,\infty)$ spaces first obtained by Garsia-Rodemich in the one dimensional case. This leads to a new proof of the John-Nirenberg self-improving inequalities. We also show a related result that provides a still a new characterization of the $L(p,\infty)$ spaces in terms of distribution functions, reflects the self-improving inequalities directly, and also characterizes $L(\infty,\infty),$ the rearrangement invariant hull of $BMO.$ We show an application to the study of tensor products with $L(\infty,\infty)$ spaces, which comp…
Using integral formulas based on Green's theorem and in particular a lemma of Uchiyama, we give simple proofs of comparisons of different BMO norms without using the John-Nirenberg inequality while we also give a simple proof of the strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA in the dual of H^1 and BMO in the dual of real H^1.
We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderon-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.
We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John-Nirenberg inequality. Moreover, we c…
A new approach to classical self improving results for $BMO$ functions is presented. "Coordinate Gagliardo spaces" are introduced and a generalized version of the John-Nirenberg Lemma is proved. Applications are provided.
We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated $BMO$ space.
We prove a local version of Fefferman-Stein inequality for the local sharp maximal function, and a local version of John-Nirenberg inequality for locally BMO functions, in the framework of locally homogeneous spaces, in the sense of Bramanti-Zhu [Manuscripta Math. 138 (2012), no. 3-4, 477-528].
Let $p\in (1,\infty)$, $q\in[1,\infty)$, $α\in [0,\infty)$ and $s$ be a non-negative integer. In this article, the authors introduce the John--Nirenberg-Campanato space $JN_{(p,q,s)_α}(\mathcal{X})$, where $\mathcal{X}$ is ${\mathbb R}^n$ or any closed cube $Q_0\subsetneqq{\mathbb R}^n$, which when $α=0$ and $s=0$ coincides with the $JN_p$-space introduced by F. John and L. Nirenberg in the sense of equivalent norms. The authors then give the predual space of $JN_{(p,q,s)_α}(\mathcal{X})$ and a John-Nirenberg type inequality of John--Nirenberg-Campanato spaces. Moreover, the authors prove that…
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 $Q_0\subset\mathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.
To shed some light on the John--Nirenberg space, the authors in this article introduce the John--Nirenberg-$Q$ space via congruent cubes, $JNQ^α_{p,q}(\mathbb{R}^n)$, which when $p=\infty$ and $q=2$ coincides with the space $Q_α(\mathbb{R}^n)$ introduced by Essén et al. in [Indiana Univ. Math. J. 49 (2000), 575-615]. Moreover, the authors show that, for some particular indices, $JNQ^α_{p,q}(\mathbb{R}^n)$ coincides with the congruent John--Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into $JNQ^α_{p,q}(\mathbb{R}^n)$. Furthermore, the authors characterize $JN…
Let $p,q\in [1,\infty]$, $α\in{\mathbb{R}}$, and $s$ be a non-negative integer. In this article, the authors introduce a new function space $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ of John-Nirenberg-Campanato type, where $\mathcal{X}$ denotes $\mathbb{R}^n$ or any cube $Q_{0}$ of $\mathbb{R}^n$ with finite edge length. The authors give an equivalent characterization of $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case $s=0$, this new space can be equivalently characterized by both maximal functio…
In this paper, we study the John-Nirenberg inequality for BMO and the atomic decomposition for H1 of noncommutative martingales. We first establish a crude version of the column (resp. row) John-Nirenberg inequality for all 0 < p < \infty. By an extreme point property of Lp -space for 0 < p \leq 1, we then obtain a fine version of this in equality. The latter corresponds exactly to the classical John-Nirenberg inequality and enables us to obtain an exponential integrability inequality like in the classical case. These results extend and improve Junge and Musat's John-Nirenberg inequality. By d…
Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint "simple" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\cap W$ and $F\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by "simple" we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space $BMO$ of functions of bounded mean oscillatio…
Let $p\in(1,\infty)$, $q\in[1,\infty)$, $s\in{\mathbb Z}_{+}$, $α\in[0,\infty)$ and $\mathcal{X}$ be $\mathbb R^n$ or a cube $Q_0\subsetneqq\mathbb R^n$. In this article, the authors first introduce the localized John--Nirenberg--Campanato space $jn_{(p,q,s)_α}(\mathcal{X})$ and show that the localized Campanato space is the limit case of $jn_{(p,q,s)_α}(\mathcal{X})$ as $p\to\infty$. By means of local atoms and the weak-$*$ topology, the authors then introduce the localized Hardy-kind space $hk_{(p',q',s)_α}(\mathcal{X})$ which proves the predual space of $jn_{(p,q,s)_α}(\mathcal{X})$. Moreov…
Let $X$ be a metric space equipped with a metric $d$ and a nonnegative Borel measure $μ$ satisfying the doubling property and let $\{\mathcal{A}_t\}_{t>0}$, be a generalized approximations to the identity, for example $\{\mathcal{A}_t\}$ is a holomorphic semigroup $e^{-tL}$ with Gaussian upper bounds generated by an operators $L$ on $L^2(X)$. In this paper, we introduce and study the weighted BMO space BMO$_\mathcal{A}(X,w)$ associated to the the family \{\mathcal{A}_t\}$. We show that for these spaces, the weighted John-Nirenberg inequality holds and we establish an interpolation theorem in s…
In this work, we are interested to develop new directions of the famous T(1)-theorem. More precisely, we develop a general framework where we look for replacing the John-Nirenberg space BMO (in the classical result) by a new BMO_{L}, associated to a semigroup of operators (e^{-tL})_{t>0}. These new spaces BMO_L (including BMO) have recently appeared in numerous works in order to extend the theory of Hardy and BMO space to more general situations. Then we give applications by describing boundedness for a new kind of paraproducts, built on the considered semigroup. In addition we obtain a versio…
It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω…
We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is…
Suppose that (M,d,m) is an unbounded metric measure space, which possesses two geometric properties, called "isoperimetric property" and "approximate midpoint property", and that the measure m is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure m is not globally doubling. In this paper we define an atomic Hardy space H1(m), where atoms are supported only on "small balls", and a corresponding space BMO(m) of functions of bounded mean oscillation, where the control is only on the oscillation over small b…
Let $p\in[1,\infty]$, $q\in(1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calderón--Zygmund operator $T$ on $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, the special John--Nirenberg--Campanato space via congruent cubes, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. Then the authors prove that $\widetilde T$ is bounded on $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if, for any $γ\in\mathbb{Z}_+^n$ with $|γ|\leq s$, $T^*(x^γ…
In this note two results are established for energy functionals that are given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over $Ω\subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$, the space of functions of Bounded Mean Oscillation of John & Nirenberg. A version of Taylor's theorem is first shown to be valid provided the integrand $W$ has polynomial growth. This result is then used to demonstrate that, for the Dirichlet, Neumann, and mixed problems, every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the se…
Let $\mathcal{X}$ be a space of homogenous type and $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ a growth function such that $φ(\cdot,t)$ is a Muckenhoupt weight uniformly in $t$ and $φ(x,\cdot)$ an Orlicz function of uniformly upper type 1 and lower type $p\in(0,1]$. In this article, the authors introduce a new Musielak-Orlicz BMO-type space $\mathrm{BMO}^φ_A(\mathcal{X})$ associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces $\mathrm{BMO}^φ_{A,\,\mathrm{max}}(\ma…
Given a Radon measure $μ$ on $R^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties that hold when $μ$ is doubling remain valid for the space BMO introduced in this paper, without assuming $μ$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from …
Let $p\in[1,\infty]$, $q\in[1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors first find a reasonable version $\widetilde{I}_β$ of the (generalized) fractional integral $I_β$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_β$. Then the authors prove that $\widetilde{I}_β$ is bounded from $JN_{(p,q,s)_α}^…
A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces (R^n,mu) with mu(B(x,r))<Cr^d, in which non-doubling harmonic analysis has recently been developed. It seems to be a promising framework for an abstract extension of this theory. Tolsa's space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven.
An RD-space ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in ${\mathcal X}$. Let $ρ$ be an admissible function on RD-space ${\mathcal X}$. The authors first introduce the localized spaces $\mathrm{BMO}_ρ({\mathcal X})$ and $\mathrm{BLO}_ρ({\mathcal X})$ and establish their basic properties, including the John-Nirenberg inequality for $\mathrm{BMO}_ρ({\mathcal X})$, several equivalent characterizations for $\mathrm{BLO}_ρ({\mathcal X})$, and some relations between these spaces. Then the author…
Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. Associated to $L$ are certain approximations of the identity. We call an operator $T$ a non-integral operator if compositions involving $T$ and these approximations satisfy certain weighted norm estimates. The Davies-Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on $T$ in Calderón-Zygmund theory. In this paper, we show, under the…
In this paper, we define a notion of $β$-dimensional mean oscillation of functions $u: Q_0 \subset \mathbb{R}^d \to \mathbb{R}$ which are integrable on $β$-dimensional subsets of the cube $Q_0$: \begin{align*} \|u\|_{BMO^β(Q_0)}:= \sup_{Q \subset Q_0} \inf_{c \in \mathbb{R}} \frac{1}{l(Q)^β} \int_{Q} |u-c| \;d\mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all finite subcubes $Q$ parallel to $Q_0$, $l(Q)$ is the length of the side of the cube $Q$, and $\mathcal{H}^β_\infty$ is the Hausdorff content. In the case $β=d$ we show this definition is equivalent to the classical no…
We study a function space $JN_p$ based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that $L^p\subset JN_{p}\subsetneq L^{p,\infty}$, but otherwise the structure of $JN_p$ is largely a mystery. Our first main result is the construction of a function that belongs to $JN_p$ but not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes $JN_{p}$ and $L^p$ do coincide. Our second main result describes $JN_p$ as the dual of a new Hardy kind of space $HK_{p'}$.
This paper is devoted to characterizing the analytic Campanato spaces $\mathcal{AL}_{p,η}$ (including the analytic Morrey spaces, the analytic John-Nirenberg space, and the analytic Lipschitz/Hölder spaces) on the complex unit disk $\mathbb D$ in terms of the Möbius mapping and the Littlewood-Paley form, and consequently their compositions with the analytic self-maps of $\mathbb D$.
This paper investigates the concept of harmonic functions of bounded mean oscillation, starting from John-Nirenberg's pioneering studies, under a renewed formalism, suitable for bringing out some fundamental properties inherent in it. In more detail: after a quick introduction, the second Section presents the main theorem, plus complete proof, relating to this function; in the third Section there is a suggestion on the exponential integrability (theorem and sketch of proof), while the fourth Section deals with the duality of Hardy Space $H^1$ and bounded mean oscillation, with some ideas for a…
In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincaré inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.
Let $ϕ: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $μ= \text{det}\,D^2 ϕ$ be the Monge-Ampère measure generated by $ϕ.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n: ϕ(y)<ϕ(x)+\nabla ϕ(x)\cdot(y-x)+t\}$ denote the section. If $μ$ satisfies the doubling property, Caffarelli and Gutiérrez (Trans. AMS 348:1075--1092, 1996) provided a variant of the Calderón-Zygmund decomposition and a John-Nirenberg-type inequality associated with sections. Under a stronger uniform continuity condition on $μ$, they also (Amer. J. Math. 119:423--465, 1997) proved an invar…
Let $L$ be the divergence form elliptic operator with complex bounded measurable coefficients, $ω$ the positive concave function on $(0,\infty)$ of strictly critical lower type $p_\oz\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors study the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$ and its dual space $\mathrm{BMO}_{ρ,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$. Several characterizations of $H_{ω,L}({\mathbb R}^n)$, including the molecular characterization, the Lusin-area function characte…
Let $φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb R^n)$ weight uniformly in $t$. In this article, the authors introduce the Musielak-Orlicz Campanato space ${\mathcal L}_{φ,q,s}({\mathbb R}^n)$ and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$, which in the case when $q=1$ and $s=0$ was obtained by L. D. Ky [arXiv: 1105.0486]. The authors also establish a John-Nirenberg inequality for functions in ${\mathcal L}_{φ,1,s}({\math…
We study the John-Nirenberg space $JN_p$, which is a generalization of the space of bounded mean oscillation. In this paper we construct new $JN_p$ functions, that increase the understanding of this function space. It is already known that $L^p(Q_0) \subsetneq JN_p(Q_0) \subsetneq L^{p,\infty}(Q_0)$. We show that if $|f|^{1/p} \in JN_p(Q_0)$, then $|f|^{1/q} \in JN_q(Q_0)$, where $q \geq p$, but there exists a nonnegative function $f$ such that $f^{1/p} \notin JN_p(Q_0)$ even though $f^{1/q} \in JN_q(Q_0)$, for every $q \in (p,\infty)$. We present functions in $JN_p(Q_0) \setminus VJN_p(Q_0)$ …
This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse Hölder inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean oscillation and parabolic Muckenhoupt weights is studied by applying a parabolic John--Nirenberg lemma. A Coifman--Rochberg-type characterization of …
