Let $T$ be an operator and suppose that there exists a positive constant $C$ such that $$\left(\int_I|Tf(x)|^q\, dx\right)^{1/q}\leq C\left(\int_I|f(x)|^q\, dx\right)^{1/q}$$ for every $q$ which is near enough to $1$ and for every interval $I$ in $\mathbb{R}$ and $f\in L^{\infty}(\mathbb{R})$. Then we show that $T$ maps $uL^{\infty}$ to ${\rm{BMO}}_u$.
In this note we generalize the definition of Fujii-Wilson condition providing quantitative characterizations of some interesting classes of weights, such as $A_\infty$, $A_\infty^{weak}$ and $C_p$, in terms of BMO type spaces suited to them. We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom's BMO type spaces.
We show that, if $b\in L^1(0,T;L^1_{\mathrm{loc}}(\mathbb{R}))$ has spatial derivative in the John-Nirenberg space $\mathrm{BMO}(\mathbb{R})$, then it generalizes a unique flow $ϕ(t,\cdot)$ which has an $A_\infty(\mathbb R)$ density for each time $t\in [0,T]$. Our condition on the map $b$ is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in $\mathrm{BMO}(\mathbb R)$.
In this note we investigate some conditions of Hörmander-Mihlin type in order to assure the $L^\infty$-${\textnormal{BMO}}$ boundedness for pseudo-multipliers of the harmonic oscillator. The $\textnormal{H}^1$-$L^1$ continuity for Hermite multipliers also is investigated. The final version of this paper will appear in Rev. Colombiana Mat.
In this paper, we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define Dunkl-type $BMO$ space and Riesz transforms for Dunkl transform on $L^\infty$, and prove the boundedness of Riesz transforms from $L^\infty$ to Dunkl-type $BMO$ space under the uniform boundedness assumption of Dunkl translations. The proof and the definition in Dunkl setting will be harder than in the classical case for the lack of some similar properties of Dunkl translations to that of classical translations. We will also extend the preciseness of the descripti…
In this note, we establish several interpolation inequalities in $\mathbb R^n$ in the Lebesgue spaces and Morrey spaces. By using the classical Calderon--Zygmund decomposition, we will reprove that $L^{p}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)\subset L^{q}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1\leq p<\infty$. We also reprove that there exists a constant $C(p,q,n)$ depending on $p,q,n$ such that the following inequality \begin{equation*} \|f\|_{L^q}\leq C(p,q,n)\cdot\big(\|f\|_{L^p}\big)^{p/q}\cdot\big(\|f\|_{\mathrm{BMO}}\big)^{1-p/q} \end{equation*} holds for all $f\in L^{p}…
For a bounded singular integral $T_n$ in $\mathbb{R}^n$ and a bounded singular integral $T_m$ in $\mathbb{R}^m$ we prove that $$ \| [T_n^1, [b, T_m^2]] \|_{L^p(μ) \to L^p(λ)} \lesssim_{[μ]_{A_p}, [λ]_{A_p}} \|b\|_{\operatorname{BMO}_{\textrm{prod}}(ν)}, $$ where $p \in (1,\infty)$, $μ, λ\in A_p$ and $ν:= μ^{1/p}λ^{-1/p}$. Here $T_n^1$ is $T_n$ acting on the first variable, $T_m^2$ is $T_m$ acting on the second variable, $A_p$ stands for the bi-parameter weights of $\mathbb{R}^n \times \mathbb{R}^m$ and $\operatorname{BMO}_{\textrm{prod}}(ν)$ is a weighted product BMO space.
Consider a non-doubling manifold with ends $M = \mathfrak{R}^{n}\sharp\, {\mathbb R}^{m}$ where $\mathfrak{R}^n=\mathbb{R}^n\times \mathbb{S}^{m-n}$ for $m> n \ge 3$. We say that an operator $L$ has a generalised Poisson kernel if $\sqrt{ L}$ generates a semigroup $e^{-t\sqrt{L}}$ whose kernel $p_t(x,y)$ has an upper bound similar to the kernel of $e^{-t\sqrtΔ}$ where $Δ$ is the Laplace-Beltrami operator on $M$. An example for operators with generalised Gaussian bounds is the Schrödinger operator $L = Δ+ V$ where $V$ is an arbitrary non-negative locally integrable potential. In this paper, our…
Let $K_t$ be either the heat or the Poisson kernel on $\mathbb{R}^n.$ Let $\mathcal{A}$ stand either for BMO equipped with the quadratic seminorm or for $A_p,$ $1< p\le\infty.$ We establish the following transference between the class $\mathcal{A}$ on an interval $I\subset\mathbb{R}$ and its $K$-version, $\mathcal{A}^K,$ on $\mathbb{R}^n$: If a given integral functional admits an estimate on $\mathcal{A}(I),$ then the same estimate holds for $\mathcal{A}^K(\mathbb{R}^n),$ with all Lebesgue averages replaced by $K$-averages. In particular, all such estimates are dimension-free. As an applicatio…
In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0< p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace-Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schrödinger operator on an Euclidean space, and (iii) $T = g…
Let $T$ be a bilinear Calderón-Zygmund singular integral operator and $T^*$ be its corresponding truncated maximal operator. For any $b\in\text{BMO}(\mathbb {R}^n)$ and $\vec{b}=(b_1,\ b_2)\in\text{BMO}(\mathbb {R}^n)\times\text {BMO}(\mathbb{R}^n)$, let $T^*_{b,j}$ (j=1,2), $T^*_{\vec{b}}\ $ be the commutators in the j-th entry and the iterated commutators of $T^*$, respectively. In this paper, for all $1<p_1,p_2<\infty$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, we show that $T^*_{b,j}$ and $T^*_{\vec{b}}$ are compact operators from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to $L^p(v_{\vec{w}})$, i…
We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,α}_x)$ estimates for $1<p,q<\infty$ and $0<α<1$, weighted mixed weak-type estimates for $q=1$, $L^\infty_{t}(L^p_x)-BMO_t(W^{2,p}_x)$, and $L^\infty_{t}(C^α_x)-BMO_t(C^{2,α}_x)$, and a.e.~pointwise formulas for derivatives, for solutions $u=u(t,x)$ to parabolic equations of the form $$\partial_tu-a^{ij}(t)\partial_{ij}u+u=f\quad t\in\mathbb{R},~x\in\mathbb{R}^n$$ and for the Cauchy problem $$\begin{cases} \partial_tv-a^{ij}(t)\partial_{ij}v+v=f&\hbox{for}~t >0,~x\in\mathbb{R}^n \\ v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$…
Let $(X,d,μ)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincaré inequality. Let $\mathscr{L}=\mathcal{L}+V$ be a Schrödinger operator on $X$, where $\mathcal{L}$ is a non-negative operator generalized by a Dirichlet form, and $V$ is a non-negative Muckenhoupt weight that satisfies a reverse Hölder condition $RH_q$ for some $q\ge (Q+1)/2$. We show that a solution to $(\mathscr{L}-\partial_t^2)u=0$ on $X\times \mathbb{R}_+$ satisfies the Carleson condition, $$\sup_{B(x_B,r_B)}\frac{1}{μ(B(x_B,r_B))} \int_{0}^{r_B} \int_{B(x_B,r_B)} |t\nabla u(x,t)|^2 \fr…
Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal L=-Δ+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal L,w}^α(\mathbb R^d)$ and $BLO_{L,w}^α(\mathbb R^d)$ in the Schrödinger setting. We prove that the variation operator $V_σ(\{T_t\}_{t>0})$, $σ>2$, and the oscillation operator $O(\{T_t\}_{t>0}, \{t_j\}_{j\in \mathbb Z})$, where $t_j<t_{j+1}$, $j\in \mathbb Z$, $\lim_{j\rightarrow +\infty}t_j=+\infty$ and $\lim_{j\rightarrow -\infty} t_j=0$, being $T_t=t^k\partial_t^k e^{-t\mat…
Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions
National Changhua University of Education（国立彰化师范大学）；BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Universidad del País Vasco / Euskal Herriko Unibertsitatea (UPV/EHU)（巴斯克大学）
BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
School of Information Network Security People’s Public Security University of China（中国人民公安大学信息网络安全学院）；College of Mathematics and System Science Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Mathematics and Statistics Minnan Normal University（闽南师范大学数学与统计学院）；School of Mathematical Sciences Xiamen University（厦门大学数学科学学院）
We provide an abstract estimate of the form \[ \|f-f_{Q,μ}\|_{X \left(Q,\frac{\mathrm{d} μ}{Y(Q)}\right)}\leq c(μ,Y)ψ(X)\|f\|_{\mathrm{BMO}(\mathrm{d}μ)} \] for all cubes $Q$ in $\mathbb{R}^n$ and every function $f\in \mathrm{BMO}(\mathrm{d}μ)$, where $μ$ is a doubling measure in $\mathbb{R}^n$, $Y$ is some positive functional defined on cubes, $\|\cdot \|_{X \left(Q,\frac{\mathrm{d} w}{w(Q)}\right)}$ is a sufficiently good quasi-norm and $c(μ,Y)$ and $ψ(X)$ are positive constants depending on $μ$ and $Y$, and $X$, respectively. That abstract scheme allows us to recover the sharp estimate \[ \…
A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to {\mathbb C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb R^n}$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of $T_g$ on these spaces are obtained.
