A Unified Version of Weighted Weak Type Inequality for Martingale Maximal Operators

Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the formfor martingale maximal operators f ∗ is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

Commutators of the B-maximal operator and B-maximal commutators

In this paper we consider the commutator of the B-maximal operator and the B-maximal commutator associated with the Laplace-Bessel differential operator. The boundedness of the commutator of the B-maximal operator with BMO symbols on weighted Lebesque space and weak-type inequality for the commutator of the B-maximal operator are proved.

Radial averaging operator acting on Bergman and Lebesgue spaces

It is shown that the radial averaging operator T_{\omega}(f)(z)=\frac{\int_{|z|}^{1}f\bigl{(}s\frac{z}{|z|}\bigr{)}\omega(s)% \,ds}{\widehat{\omega}(z)},\quad\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds, induced by a radial weight ω on the unit disc {\mathbb{D}} , is bounded from the weighted Bergman space {A^{p}_{\nu}} , where {0<p<\infty} and the radial weight ν satisfies {\widehat{\nu}(r)\leq C\widehat{\nu}(\frac{1+r}{2})} for all {0\leq r<1} , to {L^{p}_{\nu}} if and only if the self-improving condition \sup_{0\leq r<1}\frac{\widehat{\omega}(r)^{p}}{\int_{r}^{1}s\nu(s)\,ds}\int_{0% }^{r}\frac{t\nu(t)}{\widehat{\omega}(t)^{p}}\,dt<\infty is satisfied. Further, two characterizations of the weak-type inequality \eta(\{z\in\mathbb{D}:|T_{\omega}(f)(z)|\geq\lambda\})\lesssim\lambda^{-p}\|f% \|_{L^{p}_{\nu}}^{p},\quad\lambda>0, are established for arbitrary radial weights ω, ν and η. Moreover, differences and interrelationships between the cases {A^{p}_{\nu}\to L^{p}_{\nu}} , {L^{p}_{\nu}\to L^{p}_{\nu}} and {L^{p}_{\nu}\to L^{p,\infty}_{\nu}} are analyzed.

A weighted weak-type bound for Haar multipliers

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we havefor some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.