2000 Mathematics Subject Classification: 42B20, 42B25, 42B35Let K = [0, ∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we consider the generalized shift operator, generated by Laguerre hypergroup, by means of which the maximal function is investigated. For 1 < p ≤ ∞ the Lp(K)-boundedness and weak L1(K)-boundedness result for the maximal function is obtained.* V. Guliyev partially supported by grant of INTAS...

Given a rotation invariant measure in ${ℝ}^n$, we define the maximal operator over circular sectors. We prove that it is of strong type $(p,p)$ for $p>1$ and we give necessary and sufficient conditions on the measure for the weak type $(1,1)$ inequality. Actually we work in a more general setting containing the above and other situations.

In this paper, we study the Lp mapping properties of maximal functions with rough kernels that are related to certain class of singular integral operators. We prove that our maximal functions are bounded on Lp provided that their kernels are in L (log L)1/2(Sn-1). Moreover, we present an example showing that our size condition on the kernel is optimal. As a consequence of our result, we substantially improve previously known results on maximal functions, singular integral operators, and Parametric...

In this paper we prove the continuity of fractional integrals acting on nonhomogeneous function spaces defined on spaces of homogeneous type with finite measure. A definition of the molecules which are used in the $H^p$ theory is given. Results are proved for $L^p$, $H^p$, BMO, and Lipschitz spaces.

2000 Mathematics Subject Classification: 26A33, 42B20There is given a generalization of the Marchaud formula for one-dimensional fractional derivatives on an interval (a, b), −∞ < a < b ≤ ∞, to the multidimensional case of functions defined on a region in R^n

Multi-dimensional generalizations of the Wiener-Żelazko and Lévy-Żelazko theorems are obtained.

In ${ℝ}^n$, we prove $L^{p₁}\times ...\times L^{p_K}$ boundedness for the multilinear fractional integrals $I_{\alpha }(f₁,...,f_K)\left(x\right)=ʃf₁(x-\theta ₁y)...f_K(x-{\theta }_{K}y){\left|y\right|}^{\alpha -n}dy$ where the ${\theta }_j$’s are nonzero and distinct. We also prove multilinear versions of two inequalities for fractional integrals and a multilinear Lebesgue differentiation theorem.

A variety of results regarding multilinear singular Calderón-Zygmund integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal...

We prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of nontrivial...

We prove two pointwise estimates relating some classical maximal and singular integral operators. In particular, these estimates imply well-known rearrangement inequalities, $L_{\omega }^p$ and BLO-norm inequalities

Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where ${\nabla }_k$ is the gradient of order $k$, $ℳ$ is the Hardy-Littlewood maximal operator, and $I_z$ is the Riesz potential of order $z$, are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M(W_p^m\left({ℝ}^n\right)\to W_p^l\left({ℝ}^n\right))$ is described.