Related papers: Some obstacles in characterising the boundedness o…

One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say a theory is bounded if there is an axiomatization involving only $\forall_n$-formulas for some…

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calder\'on-Zygmund singular integral operator…

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 establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump…

Let $T$ be a Calder\'on-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,\kappa}(w)$ under appropriate conditions on the weight…

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(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}}…

Let $(S, d, \rho)$ be the affine group $\mathrm{R}^n \ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $\rho$, which is of exponential growth at infinity. In this paper, for any linear…

Let $T$ be an $L^2$-bounded operator having an $\omega$-Calder\'on--Zygmund kernel $K$ with a modulus of continuity $\omega$. If $\omega$ satisfied the Dini condition $\int_0^1\omega(t)\ud t/t<\infty$, then $T$ satisfies the $A_2$ theorem…

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in H\"ormander's sense. Thus, it contributes to the long-standing problem of…

In this paper, we give necessary conditions and sufficient conditions respectively for the boundedness of the singular integral operator on the weighted Morrey spaces. We observe the phenomenon unique to the case of Morrey spaces; the…

In this paper we develop a kind of A_p theory for Calderon-Zygmund operators in a non-homogeneous setting. Let \mu be a Borel measure on \R^d which may be non doubling. The only condition that \mu must satisfy is \mu(B(x,r))\leq Cr^n for…

In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that…

We characterize the geometrically doubling condition of a metric space in terms of the uniform $L^1$-boundedness of superaveraging operators, where uniform refers to the existence of bounds independent of the measure being considered.

We represent a bilinear Calder\'on-Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a…

We characterise the boundedness of a Toeplitz operator on the Bergman space with an L^1 symbol.We also prove that the compactness of a Toeplitz operator on the Bergman space with an L^1 symbol is completely determined by the boundary…

We consider an integral operator $\mathcal{I}$, special instances of which was studied in various contexts. Using an appropriate transformation we write this operator in terms of weighted composition operators. Then, we provide a…

We introduce a new sparse $T1$ theorem that estimates the dual pair associated with a Calderon-Zygmund operator by a sub-bilinear form supported on a sparse family of cubes. The main result in the paper improves previous sparse $T1$…

In this paper we consider two weight bump conditions for higher order commutators. Given $b$ and a Calder\'on-Zygmund operator $T$, define the commutator $T^1_bf=[T,b]f= bTf-T(bf)$, and for $m\geq 2$ define the iterated commutator $T^m_b f…

A differential operator $T$ satisfies the $L^2$-unique continuation property if every $L^2$-solution of $T$ that vanishes on an open subset vanishes identically. We study the $L^2$-unique continuation property of an operator $T$ acting on a…

For a wide family of multivariate Hausdorff operators, a new stronger condition for the boundedness of an operator from this family on the real Hardy space $H^1$ by means of atomic decomposition.