Related papers: A.E. Convergence vs Boundedness

As shown in [A1], the lowest constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here…

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain $L^p$-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply…

The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear H\"ormander class…

Let $E,F$ be exact operators (For example subspaces of the $C^*$-algebra $K(H)$ of all the compact operators on an infinite dimensional Hilbert space $H$). We study a class of bounded linear maps $u\colon E\to F^*$ which we call tracially…

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 consider the summability of one- and multi-dimensional trigonometric Fourier series. The Fej{\'e}r and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that…

We establish the full quasi-Banach range of $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \rightarrow L^p(\mathbb R)$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $\Omega$ to the…

The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction…

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for…

In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr\"{o}dinger operator $P=-\Delta+V(x)$ on $\mathbb{R}^n, n\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for…

We give a self-contained and introductory account of some basic functional analytic tools needed to understand maximal monotone operators in Hilbert spaces. We review domains of (possibly unbounded) operators, closed sets and closed…

We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. This operator appears very naturally in the theory of bilinear operators such as the bilinear…

This article establishes a bilinear embedding for second-order divergence-form operators with complex coefficients, characterized by the simultaneous presence of first-order terms and negative potentials. This work provides a further…

We prove a bilinear form sparse domination theorem that applies to many multi-scale operators beyond Calder\'on-Zygmund theory, and also establish necessary conditions. Among the applications, we cover large classes of Fourier multipliers,…

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 $T1$…

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…

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 $b=a+c$…

The $R$-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal $L^p$-regularity, $2<p<\infty$, for…

Let $\mu$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(\mu)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear…

Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p…