In this note the authors give the L2(n) boundedness of a class of parametric Marcinkiewicz integral with kernel function Ω in L log+L(Sn−1) and radial function h(|x|) ∈ l ∞ l(Lq)(+) for 1 < q ≦.

As its corollary, the Lp (n)(2 < p < ∞) boundedness of and and with Ω in L log+L (Sn-1) and h(|x|) ∈ l∞ (Lq)(+) are also obtained. Here and are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g*λ-function and the Lusin area function S, respectively.

Calderón type reproducing formulae with applications have been studied on one- and higher-dimensional Lipschitz graphs. In this note we study higher order Calderón reproducing formulae on star-shaped and non-star-shaped closed Lipschitz curves and surfaces.

We study DeLeeuw type theorems for certain multilinear operators on the Lebesgue spaces and on the Hardy spaces. As applications, on the torus we obtain an analog of Lacey—Thiele's theorem on the bilinear Hilbert transform, as well as analogies of some recent theorems on multilinear singular integrals by Kenig—Stein and by Grafakos—Torres.

In this paper we prove that if a weight w satisfies the condition, then the Lp(w) norm of a one-sided singular integral is bounded by the Lp(w) norm of the one-sided Hardy-Littlewood maximal function, for 1 < p < q < ∞.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

Maximal estimates are considered for solutions to an initial value problem for the Schrödinger equation. The initial value function is assumed to be radial in ℝn, n≥2.

Theorems 1 and 2 are known results concerning Lp–Lq estimates for certain operators wherein the point (1/p, 1/q) lies on the line of duality 1/p + 1/q = 1. In Theorems 1′ and 2′ we show that with mild additional hypotheses it is possible to prove Lp-Lq estimates for indices (1/p, 1/q) off the line of duality. Applications to Bochner-Riesz means of negative order and uniform Sobolev inequalities are given.

This paper is concerned with the behavior of certain principal-value, singular integral operators on L∞ and BMO defined over a local field. It is shown that unless the definition of these operators is changed appropriately, they may not be defined for some function in these spaces. Direct, constructive proofs of the existence and boundedness of the altered operators under certain smoothness conditions on the kernel are given.

This paper considers analogs of results on integral operators studied by Hörmander. Using the sharp function introduced by Fefferman and Stein, we prove weighted norm inequalities on kernel operators which map an Lp space into an Lq space, with q not equal to p. The techniques recover known results about fractional integral operators and apply to multiplier operators which satisfy a generalization of the Hörmander multiplier condition.

We prove that in a non-isotropic Euclidean space, homogeneous Lipschitz spaces of distributions, defined in terms of (generalized) Weierstrass integrals, can be characterized by means of higher order difference operators.

The BMO norm of f is equivalent to where Pt is the Poisson kernel. In this note, we show that Pt can be replaced by a nonnegative radial function h, which is positive in a neighbourhood of 0, with and , where h is the least decreasing radial majorant of h.

Let N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

It is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

We consider the function u whose Fourier transform is the positive part of the Fourier transform of a function f on Rn. If n ≤ 2 and f satisfies simple regularity conditions (in particular if f is in the Schwartz space Y(Rn), then u lies in L1(Rn). If n ≥ 3, then simple counterexamples exist; for example, if f(x) = |x|2 exp(-|x|2), then u does not lie in L1(Rn).

The purpose of this paper is to present a novel proof of a well-known relationship between functions in harmonic subspaces of L2(Rn) ∪ L1 (Rn) and their Fourier transforms. The proof uses a characterisation of spherical harmonics given by Hecke and a method developed by the author in a previous paper.