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

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

We investigate the $L^p$ boundedness for a class of parametric Marcinkiewicz integral operators associated to submanifolds and a class of related maximal operators under the $L{\left(logL\right)}^{\alpha }{(}^{n-1})$ condition on the kernel functions. Our results improve and extend some known results.

We prove Lp (and weighted Lp) bounds for singular integrals of the formp.v.  ∫Rn E (A(x) - A(y) / |x - y|) (Ω(x - y) / |x - y|n) f(y) dy,where E(t) = cos t if Ω is odd, and E(t) = sin t if Ω is even, and where ∇ A ∈ BMO. Even in the case that Ω is smooth, the theory of singular integrals with rough kernels plays a key role in the proof. By standard techniques, the trigonometric function E can then be replaced by a large class of smooth functions F. Some related operators are also considered. As...

Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m}f\left(x\right)=p.v.\int k(x-y)m(x+y)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in \mathbb{Z}}{2}^j{\phi }_j\left(2^{j}x\right)$ for a family of functions ${{\phi }_j}_{j\in \mathbb{Z}}$ satisfying conditions (1.1)-(1.3) given below.

In this paper the notions of uniformly upper and uniformly lower $\ell$-estimates for Banach function spaces are introduced. Further, the pair $(X,Y)$ of Banach function spaces is characterized, where $X$ and $Y$ satisfy uniformly a lower $\ell$-estimate and uniformly an upper $\ell$-estimate, respectively. The integral operator from $X$ into $Y$ of the form $$Kf\left(x\right)=\varphi \left(x\right){\int }_0^{x}k(x,y)f\left(y\right)\psi \left(y\right)\mathrm{d}y$$ is studied, where $k$, $\varphi$, $\psi$ are prescribed functions under some local integrability conditions, the kernel $k$ is non-negative and is assumed to satisfy certain additional...

For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^q,l^p)}^{\alpha }$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{\alpha }$. We study the closure ${(L^q,l^p)}_{c,0}^{\alpha }$ in ${(L^q,l^p)}^{\alpha }$ of the space of test functions (infinitely differentiable and with compact support in ${ℝ}^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W¹\left({(L^q,l^p)}^{\alpha }\right)$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space $W^{1,\alpha }$) and obtain...