Related papers: Compact $T(1)$ theorem \`a la Stein

We prove compactness results and characterizations for the bi-commutator $[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calder\'on-Zygmund singular integral operators $T_1, T_2$. Our strategy for proving sufficient conditions for…

Let $\mathcal{T}^*$ be the semi-group maximal function associated to the Schr\"odinger operator $-\Delta+V(x)$ with $V$ satisfying an appropriate reverse H\"{o}lder inequality. In this paper, we show that the commutator of $\mathcal{T}^*$…

Let $T$ be a bilinear Calder\'on-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…

We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of…

We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.

We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of…

We study the two-weighted off-diagonal compactness of commutators of rough singular integral operators $T_\Omega$ that are associated with a kernel $\Omega\in L^q(\mathbb{S}^{d-1})$. We establish a characterisation of compactness of the…

The commutators of bilinear Calder\'on-Zygmund operators and point-wise multiplication with a symbol in $cmo$ are bilinear compact operators on product of Lebesgue spaces. This work shows that, for certain non-degenerate Calder\'on-Zygmund…

A version of Arzel\`a-Ascoli theorem for $X$ being $\sigma$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and…

Given a Lipschitz domain $D\subset \mathbb{R}^d,$ a Calder\'on-Zygmund operator $T$ and a modulus of continuity $\omega(x),$ we solve a problem when the restricted operator $T_Df=T(f\chi_D)\chi_D$ sends the Campanato space…

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calder\'on--Zygmund operator $T$, and a pair of weights $ \sigma , \omega \in A_p$, the commutator $ [T, b]$ is compact from $ L ^{p}…

We give again a proof of non-homogeneous T1 theorem. Our proof consists of three main parts: a construction of a random dyadic lattice; an estimate of matrix coefficients of a Calder\'on--Zygmund operator with respect to random Haar basis…

Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a…

Let $1<q<p<\infty$, $\frac1r:=\frac1q-\frac1p$, and $T$ be a non-degenerate Calder\'on--Zygmund operator. We show that the commutator $[b,T]$ is compact from $L^p({\mathbb R}^n)$ to $L^q({\mathbb R}^n)$ if and only if the symbol $b=a+c$…

We prove certain two weight bump conditions are sufficient for the compactness of the commutator $[b,T]$ where $b\in CMO$ and $T$ is a Calder\'on- Zygmund operator. This is the first result for compactness in the two weight setting without…

Let $T_1$, $T_2$ be two Calder\'on-Zygmund operators and $T_{1,\,b}$ be the commutator of $T_1$ with symbol $b\in {\rm BMO}(\mathbb{R}^n)$. In this paper, the author prove that, the composite operator $T_1T_2$ satisfies the following…

We represent a general bilinear Calder\'on-Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so called sparse T1…

A new characterization of CMO(R^n) is established by the local mean oscillation. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order…

We study an alternative definition of the $H_1$-space associated to a semicommutative von Neumann algebra $L_\infty(\mathbb{R}) \overline{\otimes} \mathcal{M}$, first studied by Mei. We identify a "new" description for atoms in $H_1$. We…

This note contains a Stone-style representation theorem for compact Hausdorff spaces.