Soit $p$ une fonction polynôme de ${\mathbf{R}}^m$ dans $\mathbf{R}$. On considère la mesure ${\mu }_p$ sur le graphe de $p$ dont la projection sur ${\mathbf{R}}^m$ est la mesure de Lebesgue. On étudie ici le comportement de la transformée de Fourier ${\widehat{\mu }}_p(u,v)$ lorsque $v$ approche de 0 (de telles distributions apparaissent comme caractères de représentations de groupes de Lie nilpotents). On étend des résultats de L. Corwin et F.P. Greenleaf (Comm. on Pure and Applied Math., 31 (1975), 681–705) au cas où le gradient de la partie de $p$ homogène de plus haut degré...

We obtain sharp power-weighted $L^p$, weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.

Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf\left(x\right)={ʃ}_{a\left(x\right)}^{b\left(x\right)}f\left(t\right)dt$ with a and b certain non-negative functions are bounded from $L_u^p(0,\infty )$ to $L_v^q(0,\infty )$, 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.

We prove that the elliptic maximal function maps the Sobolev space W4,eta(R2) into L4(R2) for all eta &gt; 1/6. The main ingredients of the proof are an analysis of the intersectiQn properties of elliptic annuli and a combinatorial method of Kolasa and Wolff.

We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Let $f\in V_r\left(\right){\cup }_r\left(\right)$, where, for 1 ≤ r < ∞, $V_r\left(\right)$ (resp., ${}_r\left(\right)$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space $H_p^T$ is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the $BM{O}_q$ spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the $L_p$ norm of the sharp operator is equivalent to the $H_p^T$ norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method....

In this paper we give some criteria for the existence of compactly supported $C^{k+\alpha }$-solutions ($k$ is an integer and $0\le \alpha <1$) of matrix refinement equations. Several examples are presented to illustrate the general theory.

In this paper we deal with several characterizations of the Hardy-Sobolev spaces in the unit ball of Cn, that is, spaces of holomorphic functions in the ball whose derivatives up to a certain order belong to the classical Hardy spaces. Some of our characterizations are in terms of maximal functions, area functions or Littlewood-Paley functions involving only complex-tangential derivatives. A special case of our results is a characterization of Hp itself involving only complex-tangential derivatives....