Related papers: Two-weight estimates for sparse square functions a…
We present new results on the two-weighted boundedness of singular integral operators and $L^p$ boundedness of the Orlicz maximal function. Namely, we extend a theorem of P\'erez regarding the necessary and sufficient conditions for the…
We approach the problem of finding the sharp sufficient condition of the boundedness of all two weight Calderon--Zygmund operators. We solve this problem in $L^2$ by writing a formula for a Bellman function of the problem.
In this short note, we give a very efficient proof of a recent result of Treil-Volberg and Lacey--Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the…
In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and…
We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This…
We consider boundedness of a certain positive dyadic operator $$ T^\sigma \colon L^p(\sigma; \ \! \ell^2) \to L^p(\omega), $$ that arose during our attempts to develop a two-weight theory for the Hilbert transform in $L^p$. Boundedness of…
We continue developing the theory of conical and vertical square functions on $R^{n}$, where $\mu$ is a power bounded measure, possibly non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove…
We study the necessity of bump conditions for the boundedness of the Hardy-Littlewood maximal operator $M$ from $L^p(v)$ into $L^p(w)$, where $1<p<\infty$. The conditions in question are obtained by replacing the average of…
The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show…
We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the…
As a corollary to our main theorem we give a new proof of the result that the norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2 characteristic of a weight to the first power, a theorem of one of us. This new proof begins…
We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump…
We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb R}^n)$. In all the cases, the conditions for the boundedness are given…
The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit…
We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on $ L ^2 $ of the weights. Such a…
We study the boundedness of intrinsic square functions and their commutators on generalized Orlicz-Morrey spaces $M^{\Phi,\varphi}(\mathbb{R}^n)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type…
We study the two-weighted estimate \[ \bigg\|\sum_{k=0}^na_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\|\leq c\|f|L_{p,u}(0,\infty)\|,\tag{$*$} \] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<p\leq…
We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We find sufficient $A_p$-bump conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in BMO$ and $T$ a singular integral…
For a class of sparse operators including majorants of singular integral, square function, and fractional integral operators in a uniform manner, we prove off-diagonal two-weight estimates of mixed type in the two-weight and…
We present necessary and sufficient conditions on triples of weights $(u,v,w)$ for the boundedness of the dyadic weighted square function $S_w$ from $L^2(u)$ into $L^2(v)$. We use this characterization to obtain necessary and sufficient…