Related papers: Sharp estimates for imaginary powers of Bessel ope…

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$,…

We obtain a weak type $(1,1)$ estimate for a maximal operator associated with the classical rough homogeneous singular integrals $T_{\Omega}$. In particular, this provides a different approach to a sparse domination for $T_{\Omega}$…

Let $0<\alpha<1$ and $\frac{1}{q}=1-\alpha$. We first obtain that the function $\omega :\mathbb{Z} \rightarrow (0,\infty)$ belongs to weight class of $\mathcal{A} (1,q)(\mathbb{Z})$ if and only if discrete fractional maximal operator…

We obtain sharp estimates for the localized distribution function of M\phi, when \phi belongs to Lp,\inf where M is the dyadic maximal operator. We obtain these estimates given the L1 and Lq norm, q < p and certain weak Lp-conditions.

In this paper, we investigate the boundedness of a square function from ergodic theory on noncommutative $L_{p}$-spaces. The main result is a weak $(1,1)$ type estimate of this square function. We also show the $(L_{\infty},\mathrm{BMO})$…

The purpose of this paper is to find explicit formulas for basic objects pertaining the local potential theory of the operator $(I-\Delta)^{\alpha/2}$, $0<\alpha<2$. The potential theory of this operator is based on Bessel potentials…

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift…

Two proofs of a weighted weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for multilinear Calder\'on-Zygmund operators are given. The ideas are motivated by different proofs of the classical weak-type $(1,1)$ estimate for…

In this note the weak type estimates for fractional integrals are studied. More precisely, we adapt the arguments of Domingo-Salazar, Lacey, and Rey to obtain improvements for the endpoint weak type estimates for regular fractional sparse…

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator $(I+L)^{-s…

We get the sharp bound for weak type $(1,1)$ inequality for $n$-dimensional Hardy operator. Moreover, the precise norms of generalized Hardy operators on the type of Campanato spaces are obtained. As applications, the corresponding norms of…

In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre…

We demonstrate the $(H^1,L^{1,2})$ or $(L^p,L^{p,2})$ mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number.

We investigate weak-type $(1, 1)$ boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the…

We find the exact Bellman function associated to the level-sets of sparse operators acting on characteristic functions.

We provide sharp weak estimates for the distribution function of M\phi when on \phi we impose L1, Lq and Lp,1 restrictions. Here M is the dyadic maximal operator associated to a tree T on a non-atomic probability measure space.

For any integer $n \geq 2$, we establish $L^p(\R^n)$ inequalities for the $r$-variations of Stein-Wainger type oscillatory integral operators with general phase functions. These inequalities closely related to Carleson's theorem are sharp,…

Let $S_{\alpha}$ be the multilinear square function defined on the cone with aperture $\alpha \geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_{\alpha}$. We first obtain a sharp weighted estimate in…

In the paper we consider the Bessel differential operator L^(\mu)=\dfrac{d^2}{dx^2}+\dfrac{2\mu+1}{x}\dfrac{d}{dx} in half-line (a,\infty), a>0, and its Dirichlet heat kernel p_a^(\mu)(t,x,y). For \mu=0, by combining analytical and…

We study perturbed Bessel operators $L_{m^2}=- \partial^2_x + ( m^2 - \frac14 )\frac{1}{x^2} + Q(x)$ on $L^2]0,\infty[$, where $m\in\mathbb{C}$ and $Q$ is a complex locally integrable potential. Assuming that $Q$ is integrable near $\infty$…