On the real line, let the Fourier transform of kn be k'n(ξ) = k'(ξ-n) where k'(ξ) is a smooth compactly supported function. Consider the bilinear operators Sn(f, g)(x) = ∫ f(x+y)g(x-y)kn(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove thatΣ∞n=-∞ ||Sn(f,g)||22 ≤ C2||f||p2||g||q2.The constant C depends only upon k.

We introduce a new “weak” BMO-regularity condition for couples (X,Y) of lattices of measurable functions on the circle (Definition 3, Section 9), describe it in terms of the lattice $X^{1/2}{\left(Y^{\text{'}}\right)}^{1/2}$, and prove that this condition still ensures “good” interpolation for the couple $(X_A,Y_A)$ of the Hardy-type spaces corresponding to X and Y (Theorem 1, Section 9). Also, we present a neat version of Pisier’s approach to interpolation of Hardy-type subspaces (Theorem 2, Section 13). These two main results of the paper are...

It is known that the weak type (1,1) for the Hardy-Littlewood maximal operator can be obtained from the weak type (1,1) over Dirac deltas. This theorem is due to M. de Guzmán. In this paper, we develop a technique that allows us to prove such a theorem for operators and measure spaces in which Guzmán's technique cannot be used.

We formulate a version of the T1 theorem which enables us to treat singular integrals whose kernels need not satisfy the usual smoothness conditions. We also prove a weighted version. As an application of the general theory, we consider a class of multilinear singular integrals in ${ℝ}^n$ related to the first Calderón commutator, but with a kernel which is far less regular.

In this paper we give a sufficient condition on the pair of weights $(w,v)$ for the boundedness of the Weyl fractional integral $I_{\alpha }^{+}$ from $L^p\left(v\right)$ into $L^p\left(w\right)$. Under some restrictions on $w$ and $v$, this condition is also necessary. Besides, it allows us to show that for any $p:1\le p<\infty$ there exist non-trivial weights $w$ such that $I_{\alpha }^{+}$ is bounded from $L^p\left(w\right)$ into itself, even in the case $\alpha >1$.

In this paper we study distribution and continuity properties of functions satisfying a vanishing mean oscillation property with a lag mapping on a space of homogeneous type.

Let P(z,β) be the Poisson kernel in the unit disk , and let $P_{\lambda }f\left(z\right)={ʃ}_{\partial }P{(z,\phi )}^{1/2+\lambda }f\left(\phi \right)d\phi$ be the λ -Poisson integral of f, where $f\in L^p\left(\partial \right)$. We let $P_{\lambda }f$ be the normalization $P_{\lambda }f/P_{\lambda }1$. If λ >0, we know that the best (regular) regions where $P_{\lambda }f$ converges to f for a.a. points on ∂ are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of $P_{0}f$ toward f in an $L^p$ weakly tangential region, if $f\in L^p\left(\partial \right)$ and p > 1. In the present paper we will extend the result to symmetric spaces X of...

Let $C{V}_p\left(F\right)$ be the left convolution operators on $L^p\left(G\right)$ with support included in F and $M_p\left(F\right)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C{V}_p\left(F\right)$, $C{V}_p\left(F\right)/M_p\left(F\right)$ and $C{V}_p\left(F\right)/W$ are as big as they can be, namely have $l^{\infty }$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_p\left(F\right)$. Other subspaces of $C{V}_p\left(F\right)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

Differentiation of integrals of functions from the class $Lip(1,1)\left(I^2\right)$ with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in $Lip(1,1)\left(I^N\right)$, N ≥ 3, and $H_1^{\omega }\left(I^2\right)$ with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension.