Related papers: Two Weight Inequalities for Discrete Positive Oper…

We consider a two weight $L^{p}(\mu) \to L^{q}(\nu)$-inequality for well localized operators as defined and studied by F. Nazarov, S. Treil and A. Volberg when $p=q=2$. A counterexample of F. Nazarov shows that the direct analogue of these…

We give a new characterization of the two weight inequality for a vector-valued positive operator. Our characterization has a different flavor than the one of Scurry's and H\"{a}nninen's. The proof can be essentially derived from the…

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm…

We prove quantitative, one-weight, weak-type estimates for maximal operators, singular integrals, fractional maximal operators and fractional integral operators. We consider a kind of weak-type inequality that was first studied by…

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…

In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional…

We characterize the $L^p(\sigma)\to L^q(\omega)$ boundedness of positive dyadic operators of the form $ T(f\sigma)=\sum_{Q\in\mathscr{D}}\lambda_Q\int_Q f\,\mathrm{d}\sigma\cdot 1_Q, $ and the $L^{p_1}(\sigma_1)\times L^{p_2}(\sigma_2)\to…

We give new and elementary proofs of one weight norm inequalities for fractional integral operators and commutators. Our proofs are based on the machinery of dyadic grids and sparse operators used in the proof of the A2 conjecture.

Let $0<\gamma<n$ and $I_\gamma$ be the fractional integral operator of order $\gamma$, $I_{\gamma}f(x)=\int_{\mathbb R^n}|x-y|^{\gamma-n}f(y)\,dy$, and let $[b,I_\gamma]$ be the linear commutator generated by a symbol function $b$ and…

We study trace inequalities of the type $$ \| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \qquad f \in L^p(d\sigma), $$ in the ``upper triangle case'' $1 \leq q<p$ for integral operators $T_k$ with positive kernels, where $d\sigma$ and…

Let $\sigma$ and $\omega$ be locally finite Borel measures on $\mathbb{R}^d$, and let $p\in(1,\infty)$ and $q\in(0,\infty)$. We study the two-weight norm inequality $$ \lVert T(f\sigma) \rVert_{L^q(\omega)}\leq C \lVert f…

We define a generalized dyadic maximal operator involving the infinite product and discuss weighted inequalities for the operator. A formulation of the Carleson embedding theorem is proved. Our results depend heavily on a generalized…

In this paper, we discuss the Hardy inequality with bilinear operators on general metric measure spaces. We give the characterization of weights for the bilinear Hardy inequality to hold on general metric measure spaces having polar…

We prove mixed weak estimates of Sawyer type for fractional operators. More precisely, let $\mathcal{T}$ be either the maximal fractional function $M_\gamma$ or the fractional integral operator $I_\gamma$, $0<\gamma<n$, $1\leq p<n/\gamma$…

We give a straighforward proof of the two weight estimates of the generalized maximal operator under Sawyer type testing conditions. The proof relies on the Martingale Carleson Embedding Theorem.

We study weighted norm inequalities of $(1,q)$- type for $0<q<1$, $\Vert \mathbf{G} \nu \Vert_{L^q(\Omega, d \sigma)} \le C \, \Vert \nu \Vert, \quad \text{for all positive measures $\nu$ in $\Omega$},$ along with their weak-type…

We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the…

In this paper, we present a sufficient condition on a pair of nonnegative weights $v$ and $w$ such that we have a general weighted $L^{p}$-Hardy type identity. The result, for a certain choice of weights, gives weighted $L^{p}$-Hardy type…

Weighted $L^p-L^r$ inequalities with arbitrary measurable non-negative weights for positive quasilinear integral operators with Oinarov's kernel on the semiaxis are characterized. Application to the boundedness of maximal operator in the…

The paper is devoted to two-weight estimates for the fractional maximal operators $\mathcal{M}^\alpha$ on general probability spaces equipped with a tree-like structure. For given $1<p\leq q<\infty$, we study the sharp universal upper bound…