We call an $L^p$-multiplier m tame if for each complex homomorphism χ acting on the space of $L^p$ multipliers there is some ${\gamma }_0\in \Gamma$ and |a| ≤ 1 such that $\chi \left(\gamma m\right)=am\left({\gamma }_0\gamma \right)$ for all γ ∈ Γ. Examples of tame multipliers include tame measures and one-sided Riesz products. Tame multipliers show an interesting similarity to measures. Indeed we show that the only tame idempotent multipliers are measures. We obtain quantitative estimates on the size of $L^p$-improving tame multipliers which are similar to those obtained for measures, but...

Let f ∈ L∞ and g ∈ L2 be supported on [0,1]. Then the principal value integral below exists in L1.p.v.   ∫ f(x + y) g(x - y) dy / y.

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f\left(s\right)={\sum }_{n=1}^{\infty }aₙn^{-s}$ with ${\left|\right|f\left|\right|}_{\infty }:=su{p}_{\Re s>0}\left|f\left(s\right)\right|<\infty$, then ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-2}{\le \left|\right|f\left|\right|}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-1/2}{\le C\left|\right|f\left|\right|}_{\infty }$, C being an absolute constant.

Calderón-Zygmund operators are generalizations of the singular integral operators introduced by Calderón and Zygmund in the fifties [CZ]. These singular integrals are principal value convolutions of the formTf(x) = límε→0 ∫|x-y|&gt;ε K(x-y) f(y) dy = p.v.K * f(x),where f belongs to some class of test functions.

The main purpose of this paper is to prove the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space.

The boundednees of multilinear commutators of Calderón-Zygmund singular integrals on Lebesgue spaces with variable exponent is obtained. The multilinear commutators of generalized Hardy-Littlewood maximal operator are also considered.

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

We investigate the relations between the Campanato, Morrey and Hölder spaces on spaces of homogeneous type and extend the results of Campanato, Mayers, and Macías and Segovia. The results are new even for the ℝⁿ case. Let (X,d,μ) be a space of homogeneous type and (X,δ,μ) its normalized space in the sense of Macías and Segovia. We also study the relations of these function spaces for (X,d,μ) and for (X,δ,μ). Using these relations, we can show that theorems for the Campanato, Morrey or Hölder spaces...

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

In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.

The boundedness properties of commutators for operators are of central importance in Mathematical Analysis, and some of these commutators arise in a natural way from interpolation theory. Our aim is to present a general abstract method to prove the boundedness of the commutator $[T,\Omega ]$ for linear operators $T$ and certain unbounded operators $\Omega$ that appear in interpolation theory, previously known and a priori unrelated for both real and complex interpolation methods, and also to show how the abstract result...

We first show that a linear operator which is bounded on $L{²}_w$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h{¹}_w$ if and only if this operator is uniformly bounded on all $h{¹}_w$-atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h{¹}_w$.