Related papers: Imaginary powers of $(k,1)$-generalized harmonic o…

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global H\"ormander classes of order $-n(1-\rho)/2$ on $G\times \widehat{G}$. Our approach…

For polynomial $ P (x,y)$, and any Calder\'{o}n-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{\epsilon >0} \Bigl\lvert \int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (x,y) } K(y) \; dy…

We prove a T(1) Theorem to completely characterize compactness of Calderon-Zygmund operators. The result provides sufficient and necessary conditions for the compactness of singular integral operators acting on L^p(R).

We consider a class of H\"ormander-type oscillatory integral operators in $\mathbb{R}^n$ for $n \geq 3$ odd with real analytic phase. We derive weak conditions on the phase which ensure $L^p$ bounds beyond the universal $p \geq 2 \cdot…

We develop new local $T1$ theorems to characterize Calder\'on-Zygmund operators that extend boundedly or compactly on $L^{p}(\mathbb R^{n},\mu)$ with $\mu$ a measure of power growth. The results, whose proofs do not require random grids,…

In this note we address the continuity of strongly singular Calder\'on-Zygmund operators on Hardy-Morrey spaces $\mathcal{HM}_{q}^{\lambda}(\mathbb{R}^n)$, assuming weaker integral conditions on the associated kernel. Important examples…

We give a new formulation of the $T1$ theorem for compactness of Calder\'on-Zygmund singular integral operators. In particular, we prove that a Calder\'on-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in…

In this paper we investigate a class of domains $\Omega^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ for $k \in \mathbb{Z}^+$ which generalizes the Hartogs triangle. we first obtain the new explicit formulas for the…

We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L \log L on the unit annulus is bounded on L^p for 1 < p < \infty and is of weak-type (1,1), generalizing the result of Seeger. The proof is in a…

Given sparse collections of measurable sets $\mathcal S_k$, $k=1,2,\ldots ,N$, in a general measure space $(X,\mathfrak M,\mu)$, let $ \Lambda_{\mathcal S_k}$ be the sparse operator, corresponding to $\mathcal S_k$. We show that the maximal…

Let $K$ be a standard H\"older continuous Calder\'on--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed…

We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly…

We investigate the global boundedness of Fourier integral operators with amplitudes in the general H\"ormander classes $S^{m}_{\rho, \delta}(\mathbb{R}^n)$, $\rho, \delta\in [0,1]$ and non-degenerate phase functions of arbitrary rank…

We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the…

Stein and Wainger proved the $L^p$ bounds of the polynomial Carleson operator for all integer-power polynomials without linear term. In the present paper, we partially generalise this result to all fractional monomials in dimension one.…

The purpose of this article is to provide an alternative proof of the weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for $m$-multilinear Calder\'on-Zygmund operators on $\mathbb{R}^n$ first proved by Grafakos and Torres.…

For a Calderon-Zygmund operator T on d-dimensional space, that has a sufficiently smooth kernel, we prove that for any 1< p \le 2, and weight w in A_p, that the maximal truncations T_* of T map L^p(w) to weak-L^p(w), with norm bounded by…

In this paper we establish the $L^p$-$L^q$ estimates for global pseudo-differential operators on graded Lie groups. We provide both necessary and sufficient conditions for the $L^p$-$L^q$ boundedness of pseudo-differential operators…

In the present paper, we establish that Riesz transforms for Dunkl Hermite expansion as introduced in [4] are singular integral operators with H\"ormander's type conditions and we show that are bounded on $L^p(\mathbb{R}^d; d\mu_k) 1 < p <…

We study Calder\'on-type commutators $[M_b,T_i\mathcal R_j]$ in the rational Dunkl setting with a finite reflection group $G$. If $b$ belongs to the orbit Lipschitz class $\operatorname{Lip}_d$, then for every $1<p<\infty$ we prove…