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With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of…

We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space…

We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities of families. The most general form of this compactness notion depends on two ordinal parameters. In the…

The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper we focus on the mapping properties of the volume…

Given a positive definite kernel in a locally compact space, we study a minimal energy problem in the presence of an external field over the class of all nonnegative Radon measures that are supported by a given closed noncompact set,…

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 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…

In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as…

Let $T_\Omega$ be the singular integral operator with variable kernel $\Omega(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_\Omega$ on these…

We demonstrate and develop dyadic-probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a…

We introduce a new combinatorial condition that characterises the amenability for locally compact groups. Our condition is weaker than the well-known F{\o}lner's conditions, and so is potentially useful as a criteria to show the amenability…

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation…

We investigate the stability of compactness of bilinear operators acting on the product of interpolation of Banach spaces. We develop a general framework for such results and our method applies to abstract methods of interpolation in the…

In this paper we prove an abstract homomorphism theorem for bilinear multipliers in the setting of locally compact Abelian (LCA) groups. We also provide some applications. In particular, we obtain a bilinear abstract version of K. de…

We prove necessary conditions on pairs of measures $(\mu,\nu)$ for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p<\infty$, provided the kernel of $T$ satisfies a weak non-degeneracy condition first…

Let $T_1$, $T_2$ be two singular integral operators with nonsmooth kernels introduced by Duong and McIntosh. In this paper, by establishing certain bi-sublinear sparse domination, the authors obtain some quantitative bounds on…

In the framework of computational complexity and in an effort to define a more natural reduction for problems of equivalence, we investigate the recently introduced kernel reduction, a reduction that operates on each element of a pair…

The optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of singular integral operators of both Calder\'on-Zygmund and of rough type are shown in the different exponent ranges $``q>p"$, $``q=p"$ and $``q<p"$ to…

We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of…

The paper studies strictly positive definite kernels on compact Riemannian manifolds. We state new conditions to ensure strict positive definiteness for general kernels and kernels with certain convolutional structure. We also state…