R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on $L^2$ maps $L^{\infty }$ into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by $C\left|R{|}^s\right|R{|}^{-s}$, for some $s\>0$. We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided $s\>s_0\left(E\right)$. We also show that the Calderon-Coifman bicommutators,...

We give some explicit values of the constants $C_1$ and $C_2$ in the inequality $C_1/sin\left(\frac{\pi }{p}\right)\le {\left|P\right|}_p\le C_2/sin\left(\frac{\pi }{p}\right)$ where ${\left|P\right|}_p$ denotes the norm of the Bergman projection on the $L^p$ space.

The purpose of this paper is to introduce some new generalized double difference sequence spaces using summability with respect to a two valued measure and an Orlicz function in $2$-normed spaces which have unique non-linear structure and to examine some of their properties. This approach has not been used in any context before.

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 the present paper we define and study the properties of a deformation of measures and convolutions that works in a similar way to the $U_t$ deformation of Bożejko and Wysoczański, but in its definition operates on two levels of Jacobi coefficients of a measure, rather than on one.

In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and $q$-deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a $*$-algebra equipped with a state) drawn from an ensemble of pair-wise commuting...

In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong $A_{\mathrm{\infty }}$ weight with respect to the metric associated with the operator. Roughly speaking, the strong $A_{\mathrm{\infty }}$ condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....

Necessary and sufficient conditions are shown in order that the inequalities of the form $\varrho \left(M_{\mu }f>\lambda \right)\Phi \left(\lambda \right)\le C{ʃ}_X\Psi \left(C\right|f\left(x\right)\left|\right)\sigma \left(x\right)d\mu$, or $\varrho \left(M_{\mu }f>\lambda \right)\le C{ʃ}_X\Phi (C{\lambda }^{-1}|f\left(x\right)\left|\right)\sigma \left(x\right)d\mu$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_{\mu }$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.