Related papers: Quantitative BMO-BLO Estimates for the Hardy-Littl…

In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times…

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions…

We find the best possible constant $C$ in the inequality $\|\varphi\|_{L^r}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$, where $2 \leq r$ and $p < r$. We employ the Bellman function technique to solve…

We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on…

We prove a local $Tb$ theorem under close to minimal (up to certain `buffering') integrability assumptions, conjectured by S. Hofmann (El Escorial, 2008): Every cube is assumed to support two non-degenerate functions $b^1_Q\in L^p$ and…

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_\rho$ and phases $\varphi$ such that $\varphi(x,\xi) -…

For $\alpha\in(0, n)$ and a growth function $\varphi:[0,\infty)\rightarrow [0,\infty)$, it is proved that the commutator $[b,I_\alpha]$ generated by fractional integral operator $I_\alpha$ and Orlicz $\mathrm{BMO}$ function $b$ is bounded…

A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, where $f$ is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and…

Let $\{A_t\}_{t>0}$ be the dilation group in ${\Bbb R}^n$ generated by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let $\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$ be a $A_t$-homogeneous distance function defined on…

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that…

In \cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \cite{AP2} for certain special classes of functions. In particular, we…

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form $m(\xi,\eta)=e^{i \phi(\xi-\eta)}$. We prove that if $\phi$ is a $C^1(\mathbb R^n)$ real-valued…

Let $\mathcal{E}(X,d,\mu)$ be a Banach function space over a space of homogeneous type $(X,d,\mu)$. We show that if the Hardy-Littlewood maximal operator $M$ is bounded on the space $\mathcal{E}(X,d,\mu)$, then its boundedness on the…

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing…

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…

In this paper we study the boundedness on $L^p(w)$ of the maximal operator $M_{A^{-1}}$, defined by $M_{A^{-1}}f(x)=Mf(A^{-1}x)$, that is, the maximal of Hardy-Littlewood composed with a invertible matrix $A$. We present two different…

We establish weighted inequalities for $BMO$ commutators of sublinear operators for all $0<p<\infty$. For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse H\"{o}lder condition, we prove that $\bullet$…

We extend Stein's maximal theorem to the bilinear setting. Let $M$ be a homogeneous space with a transitive action of a compact abelian group, and let $1 \le p,q \le 2$ and $1/2 \le r \le 1$ satisfy $1/p + 1/q = 1/r$. For a family of…

We prove that one can extend any $BMO^{x}$ function $a$ given in a cube in $\mathbb{R}^{d+1}$ to become a $BMO^{x}$ functions $\hat a$ in $\mathbb{R}^{d+1}$ almost preserving its $[a]^{\sharp}$ seminorm, which is, loosely speaking,…

This note contains two simple observations. First, by the weak factorization of product $H^1$ (Ferguson--Lacey, Lacey--Terwilleger), we obtain a multi-parameter analogue of Hardy's inequality. Second, as a dual statement, the Fourier…