Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the...

In this paper, we prove some weighted inequalities for the multilinear operators related to certain integral operators on the generalized Morrey spaces by using the sharp estimates of the multilinear operators. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

We prove sharp weighted inequalities of the form$${\int }_{{\mathbf{R}}^n}|f\left(x\right){|}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}|q\left(D\right)\left(f\right)\left(x\right){|}^{p}N\left(v\right)\left(x\right)dx$$where $q\left(D\right)$ is a differential operator and $N$ is a combination of maximal type operator related to $q\left(D\right)$ and to $p$.

For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf\left(x\right)=1/\left|B\right(0,\left|x\right|\left)\right|{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$, $Tf\left(x\right)=1/\left|B\right(x,\left|x\right|\left)\right|{\int }_{B(x,|x\left|\right)}f\left(t\right)dt$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

In the context of the spaces of homogeneous type, given a family of operators that look like approximations of the identity, new sharp maximal functions are considered. We prove a good-λ inequality for Muckenhoupt weights, which leads to an analog of the Fefferman-Stein estimate for the classical sharp maximal function. As a consequence, we establish weighted norm estimates for certain singular integrals, defined on irregular domains, with Hörmander conditions replaced by some estimates which do...

We investigate the boundedness of the fractional maximal operator with respect to a general basis on weighted Lebesgue spaces. We characterize the boundedness of these operators for one-weight and two-weight inequalities extending the work of Jawerth. A new two-weight testing condition for the fractional maximal operator on a general basis is introduced extending the work of Sawyer for the basis of cubes. We also find the sharp dependence in the two-weight case between the operator norm and the...

We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let $H_L^p\left(X\right)$ (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on $H_L^p\left(X\right)$ follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on $H_L^p\left(X\right)$.

Si presentano alcuni risultati recenti riguardanti la disuguaglianza di Pòlya- Szegö e la caratterizzazione dei casi in cui essa si riduce ad un'uguaglianza. Particolare attenzione viene rivolta alla simmetrizzazione di Steiner di insiemi di perimetro finito e di funzioni di Sobolev.

In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral operators of the form aP + bQ, where a,b are piecewise continuous functions and P,Q are complementary projections associated to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves.

Let χ be a space of homogeneous type. The aims of this paper are as follows.i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(χ) for 1 < p ≤ 2; our condition is weaker then the usual Hörmander integral condition.ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type...

We examine several scalar oscillatory singular integrals involving a real-analytic phase function φ(s,t) of two real variables and illustrate how one can use the Newton diagram of φ to efficiently analyse these objects. We use these results to bound certain singular integral operators.