For two general second order parabolic equations in divergence form in Lip(1,1/2) cylinders, we give a criterion for the preservation of $L^p$ solvability of the Dirichlet problems.

For an L²-bounded Calderón-Zygmund Operator T acting on $L²\left({ℝ}^d\right)$, and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_T{\left|\right|w\left|\right|}_{A₂}$. The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can be exactly...

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${\left|\right|Mf\left|\right|}_{p),w}{\le c\left|\right|f\left|\right|}_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_p$, and therefore iff ${\left|\right|Mf\left|\right|}_{p,w}{\le c\left|\right|f\left|\right|}_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....

We give some rather weak sufficient condition for $L^p$ boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ on the product spaces $ℝⁿ\times {ℝ}^m$ (1 < p < ∞), which improves and extends some known results.

We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f\left(x\right)=in{f}_{Q\ni x}1/\left|Q\right|{\int }_Q\left|f\right|dy$, extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ℝⁿ weighted norm inequalities for the geometric maximal operator $M₀f\left(x\right)={sup}_{Q\ni x}exp(1/|Q|{\int }_{Q}log\left|f\right|dx)$, proved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal to the closely...

We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt $A_p$ class.

It is shown that the Muckenhoupt structure constants for f and f* on the real line are the same.

We introduce the one-sided minimal operator, $m^{+}f$, which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided $\left(A_p^{+}\right)$ weights.

Kato’s conjecture, stating that the domain of the square root of any accretive operator $L=-div\left(A\nabla \right)$ with bounded measurable coefficients in ${ℝ}^n$ is the Sobolev space $H^1\left({ℝ}^n\right)$, i.e. the domain of the underlying sesquilinear form, has recently been obtained by Auscher, Hofmann, Lacey, McIntosh and the author. These notes present the result and explain the strategy of proof.

Figà-Talamanca characterized the space of Fourier multipliers as the dual space of a certain Banach space. In this paper, we characterize the space of maximal Fourier multipliers as a dual space.

2000 Math. Subject Classification: Primary 42B20, 42B25, 42B35In this paper we study the Riesz potentials (B -Riesz potentials) generated by the Laplace-Bessel differential operator ∆B.* Akif Gadjiev’s research is partially supported by the grant of INTAS (project 06-1000017-8792) and Vagif Guliyev’s research is partially supported by the grant of the Azerbaijan–U.S. Bilateral Grants Program II (project ANSF Award / 16071) and by the grant of INTAS (project 05-1000008-8157).

Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $\mu \left(E\right)={\int }_D{\chi }_E(x,\phi \left(x\right))dx$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_{\mu }$ be the convolution operator with the measure μ. Let $\phi =\phi {₁}^{e₁}\cdots \phi {ₙ}^{eₙ}$ be the decomposition of φ into irreducible factors. We show that if $e_i\ne m/2$ for each ${\phi }_i$ of degree 1, then the type set $E_{\mu }:=(1/p,1/q)\in [0,1]\times [0,1]:\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty$ can be explicitly described as a closed polygonal region.