A classical result states that the Hardy--Littlewood maximal operator is bounded on an Orlicz space $L^A(\mathbb{R}^n)$ if and only if its conjugate Young function $\tilde{A}$ satisfies the $\Delta_2$-condition. The same condition also characterizes the boundedness on $L^A(0,\infty)$ of the Hardy averaging operator. We consider a discrete analogue of the problem, extended to a general interpolation framework. We offer several characterizing conditions for the boundedness of discrete maximal and average operators on Orlicz spaces. Although the principal result is as expected, for its proof some new techniques have to be developed. To this end, we introduce a new notion of the so-called $\Delta_2$-fundamental sequence, and give its interesting characterization by a simple condition involving only a limes superior of the ratio of two subsequent terms. We also prove a dual statement concerning operators of Copson type.
The purpose of this note is to establish further properties of the variable Lorentz spaces $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$ introduced by L. Ephremidze, V. Kokilashvili and S. Samko, which will allow us to apply the theory of Hardy spaces associated with ball quasi-Banach function spaces, and so define the variable Hardy-Lorentz spaces associated with $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$. Then, the finite and infinite atomic decompositions for these spaces will be deduced immediately. We also obtain the boundedness of singular integrals and fractional type operators in variable Hardy-Lorentz spaces, and provide a Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $r$-convexification of variable Lorentz spaces.
The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0<p<\infty$, in terms of the Littlewood-Paley g-function $$ S f (\zeta) = \left(\int_0^1 |f'(r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X$ on the unit circle $\mathbb T$ under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded in $(X^\delta)'$ with some $\delta>0$. As an application to composition operators $C_\varphi$, we derive an exact criterion for the boundedness and compactness of $C_\varphi : \mathcal{B}^\omega \to X_A$, where $\mathcal{B}^\omega = \{f \mid \sup |f'|/\omega<\infty\}$ is the weighted Bloch space with a log-convex radial weight $\omega$, generalizing recent results in the one-dimensional setting.
E. Abakumov, Evgueni Doubtsov, D. Rutsky· 0 citations
Let $A\in\mathcal{B}(H)^+$ and $B\in\mathcal{B}(\ell^2)^+$ be positive bounded operators. We investigate reduced weighted adjoints of densely defined closable operators and use them to develop frame-type systems in the semi-Hilbert spaces induced by $A$ and $B$. Closedness, boundedness, range behavior, and algebraic properties of the $A$-adjoint are established, with particular attention to sums, products, and double adjoints in the unbounded setting. We then study the associated $AB$-analysis, $AB$-synthesis, and $AB$-frame operators. Operator-theoretic characterizations of $AB$-Bessel sequences, $AB$-lower semi-frames, and $AB$-frames are obtained, and examples show that several natural weighted-adjoint identities may hold only as proper inclusions. A Douglas-type range characterization for $AB$-frames is also derived.
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local H\"older continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky--Spencer and Bourgain--Schlag for Schr\"odinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $\gamma\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(\mu)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(\mu)} \sim \|\rho^{-1}\phi^{-\gamma}F\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}, \qquad F\in\{\Delta f,m_f\},\quad \phi\in\{\rho,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(\mu)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu(B(x,\rho(x,y)))$ and $\Delta f(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(\mu)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma\in(-\infty,-1)\cup(0,\infty)$ for $\phi=\rho$ or $\gamma\in (-\infty,-\frac1d)\cup(0,\infty)$ for $\phi=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.
Tuomas Hytönen, Dachun Yang, Wen Yuan et al.· 0 citations
We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1<p<\infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available, the global $L^p$ setting has neither. We control the non-local tail through Calder\'on--Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncation, obtaining order-$(-\epsilon)$ smoothing of the remainder into the Besov scale. Pairing weakly convergent sequences with their canonical Nemyckij duals, we then use the lemma to derive $L^p$ transport equations. For first-order scalar equations we establish the phase-space bicharacteristic (Vlasov) flow of the associated H-distribution when $p \ge 2$; for the quasilinear $p$-wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core $p = 2$ and isolating the structural obstruction to its closure when $p \neq 2$. These results provide functional-analytic tools for tracking the propagation of singularities in degenerate nonlinear and non-local partial differential equations.