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.

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....

Let ω be a Békollé-Bonami weight. We give a complete characterization of the positive measures μ such that ${\int }_{}|M_{\omega }{f\left(z\right)|}^{q}d\mu \left(z\right)\le C({\int }_{}{\left|f\left(z\right)\right|}^p{\omega \left(z\right)dV\left(z\right))}^{q/p}$ and $\mu \left(z\in :Mf\left(z\right)>\lambda \right)\le C/\left({\lambda }^q\right)({\int }_{}|f\left(z\right){{|}^p\omega \left(z\right)dV\left(z\right))}^{q/p}$, where $M_{\omega }$ is the weighted Hardy-Littlewood maximal function on the upper half-plane and 1 ≤ p,q <; ∞.

We define Beurling-Orlicz spaces, weak Beurling-Orlicz spaces, Herz-Orlicz spaces, weak Herz-Orlicz spaces, central Morrey-Orlicz spaces and weak central Morrey-Orlicz spaces. Moreover, the strong-type and weak-type estimates of the Hardy-Littlewood maximal function on these spaces are investigated.

The famous result of Muckenhoupt on the connection between weights w in Ap-classes and the boundedness of the maximal operator in Lp(w) is extended to the case p = ∞ by the introduction of the geometrical maximal operator. Estimates of the norm of the maximal operators are given in terms of the Ap-constants. The equality of two differently defined A∞-constants is proved. Thereby an answer is given to a question posed by R. Johnson. For non-increasing functions on the positive real line a parallel...