We discuss recent progress on issues surrounding the Brascamp–Lieb inequalities.

In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s > 2 to obtain local solutions for any initial data, and global solutions for small initial data.

In this work we give sufficient and necessary conditions for the boundedness of the fractional integral operator acting between weighted Orlicz spaces and suitable $BM{O}_{\phi }$ spaces, in the general setting of spaces of homogeneous type. This result generalizes those contained in [P1] and [P2] about the boundedness of the same operator acting between weighted $L^p$ and Lipschitz integral spaces on ${ℝ}^n$. We also give some properties of the classes of pairs of weights appearing in connection with this boundedness.

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:=d^2/d{x}^2-2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1\left(d\gamma \right)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.

In this paper we study the Hilbert transform and maximal function related to a curve in R2.

A version of the John-Nirenberg inequality suitable for the functions $b\in \mathrm{BMO}$ with $b^{-}\in L^{\infty }$ is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.

X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on ${ℝ}^d$. This new BMO space is very suitable for the Calderón-Zygmund theory with non-doubling measures. Especially, the John-Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces.

For two general second order parabolic equations in divergence form in Lip(1,1/2) cylinders, we give a criterion for the preservation of $L^p$ solvability of the Dirichlet problems.

For an L²-bounded Calderón-Zygmund Operator T acting on $L²\left({ℝ}^d\right)$, and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_T{\left|\right|w\left|\right|}_{A₂}$. The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can be exactly...

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${\left|\right|Mf\left|\right|}_{p),w}{\le c\left|\right|f\left|\right|}_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_p$, and therefore iff ${\left|\right|Mf\left|\right|}_{p,w}{\le c\left|\right|f\left|\right|}_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....

We give some rather weak sufficient condition for $L^p$ boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ on the product spaces $ℝⁿ\times {ℝ}^m$ (1 < p < ∞), which improves and extends some known results.