New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s < n, [resp. 0 < s < n] sends the weighted Lebesgue space Lp(v(x)dx) into Lp(u(x)dx), 1 < p < ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)≔v\left(x\right){ʃ}_0^{\infty }u\left(t\right)f\left(t\right)dt$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

We consider the Volterra integral operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)=v\left(x\right){ʃ}_0^{x}u\left(t\right)f\left(t\right)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n\left(T\right)$ of T are established when 1 < p < ∞. When p = 2 these yield $li{m}_{n\to \infty }n{a}_n\left(T\right)={\pi }^{-1}{ʃ}_0^{\infty }\left|u\left(t\right)v\left(t\right)\right|dt$. We also provide upper and lower estimates for the ${\ell }^{\alpha }$ and weak ${\ell }^{\alpha }$ norms of (an(T)) when 1 < α < ∞.

Necessary and sufficient conditions governing two-weight $L^p$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential...

We prove weighted inequalities for commutators of one-sided singular integrals (given by a Calder’on-Zygmund kernel with support in $(-\infty ,0)$) with BMO functions. We give the one-sided version of the results in C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl., vol. 3 (6), 1997, pages 743–756 and C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., vol 128 (1), 1995, pages...

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ to hold when ${\Phi }_1$ and ${\Phi }_2$ are N-functions with ${\Phi }_2\circ {\Phi }_1^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.