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We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and…

Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density…

In this paper, we give a characterization of all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral operator in $L_2(R)$ with bounded and arbitrarily smooth Carleman kernel on $R^2$. In…

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result…

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously…

Let $\mathbb{G}$ be any Carnot group. We prove that if a convolution type singular integral associated with a $1$-dimensional Calder\'on-Zygmund kernel is $L^2$-bounded on horizontal lines, with uniform bounds, then it is bounded in $L^p, p…

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of wave operators $W_\pm$ of scattering theory for Schr\"odinger operator $H=-\lap + V$ on $\R^m$, $|V(x)|\leq C\ax^{-\delta}$ for some $\delta>2$ when $H$ is of exceptional…

Consider the maximal operator $$\mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\Big|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e(\lambda |y|^{2d}) K(y)\Big|,\quad (x\in\mathbb{Z}^n),$$ where $d$ is a positive integer, $K$ a…

We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ, Christ-Rubio de Francia, and Duoandikoetxea-Rubio de Francia gives rise to maximally…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when…

We reduce the boundedness of operators in Morrey spaces $L_p^r({\mathbb R}^n)$, its preduals, $H^{\varrho}L_p ({\mathbb R}^n)$, and their preduals $\overset{\circ}{L}{}^r_p({\mathbb R}^n)$ to the boundedness of the appropriate operators in…

This paper investigates the $L^p$-bounds of wave operators for higher-order Schr\"odinger operators $H = (-\Delta)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the…

$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal…

The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size…

For $n \geq 2$, we consider the operator $L_A = -\mathrm{div }(A(\cdot)\nabla)$, where $A$ is a uniformly elliptic $(n+1)\times(n+1)$ matrix with variable coefficients, a Radon measure $\mu$ on $\mathbb{R}^{n+1}$, and the associated…

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,L_n on a Lie group G of polynomial growth, which generate…

Let $K: \boldsymbol{\Omega}\times \boldsymbol{\Omega}$ be a continuous Mercer kernel defined on a compact subset of ${\mathbb R}^n$ and $\mathcal{H}_K$ be the reproducing kernel Hilbert space (RKHS) associated with $K$. Given a finite…

Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. \int_{\mathbb{R}^{n}}K(x,x-y)f(y)\mathrm{d}y$ and $D^{\gamma}(0\leq\gamma\leq1)$ be the fractional differentiation operator, where…

Let $\lambda$ be a complex number in the closed unit disc $\overline{\Bbb D}$, and $\cal H$ be a separable Hilbert space with the orthonormal basis, say, ${\cal E}=\{e_n:n=0,1,2,\cdots\}$. A bounded operator $T$ on $\cal H$ is called a…

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…