Two-weight Norm Inequalities Research Articles

Two-weight norm inequalities for rough fractional integral operators on Morrey spaces

We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces.

Two-weight Norm Inequalities for Local Fractional Integrals on Gaussian Measure Spaces

In this paper, the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights. More precisely, the authors first obtain the two-weight weak-type estimate for the local-$a$ fractional maximal operators of order $\alpha$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with $1\leq p\leq q<\infty$ under a condition of $(u,v)\in \bigcup_{b'>a} A_{p,q,\alpha}^{b'}$, and then obtain the two-weight weak-type estimate for the local fractional integrals. In addition, the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ and the two-weight strong-type boundedness of the local fractional integrals. These estimates are established by the radialization method and dyadic approach.

Two-weight norm inequalities for some fractional type operators related to Schrödinger operator on weighted Morrey spaces

Two-weight norm inequalities for some fractional type operators related to Schrödinger operator on weighted Morrey spaces

Two-weight norm inequalities for product fractional integral operators

In the open range 1<p<q<∞, under a certain geometrical condition on weights, two-weight norm inequality for the product fractional integral operator, whose kernel has singularity appeared on each of the coordinate subspaces, is shown to follow from the Fefferman–Phong-type characteristic for the product cubes. This geometrical condition turns out to be the testing condition of Carleson-type embedding for product dyadic cubes.

On Two-Weight Norm Inequalities for Positive Dyadic Operators

Let σ and ω be locally finite Borel measures on ℝd, and let pin (1,infty ) and qin (0,infty ). We study the two-weight norm inequality lVert T(fsigma ) rVert _{L^{q}(omega )}leq C lVert f rVert _{L^{p}(sigma )}, quad text {for all} f in L^{p}(sigma ), for both the positive summation operators T = Tλ(⋅σ) and positive maximal operators T = Mλ(⋅σ). Here, for a family {λQ} of non-negative reals indexed by the dyadic cubes Q, these operators are defined by T_{lambda }(fsigma ):={sum }_{Q} lambda _{Q} langle {f}rangle ^{sigma }_{Q} 1_{Q} quad text { and } quad M_{lambda }(fsigma ):=sup _{Q} lambda _{Q} langle {f}rangle ^{sigma }_{Q} 1_{Q}, where langle {f}rangle ^{sigma }_{Q}:=frac {1}{sigma (Q)} {int limits }_{Q} |f| d sigma . We obtain new characterizations of the two-weight norm inequalities in the following cases: (1) For T = Tλ(⋅σ) in the subrange q < p. Under the additional assumption that σ satisfies the A_{infty } condition with respect to ω, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. (2) For T = Mλ(⋅σ) in the subrange q < p. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap. (3) For the summation operators T = Tλ(⋅σ) in the subrange 1 < q < p. We characterize the inequality for summation operators by means of related inequalities for maximal operators T = Mλ(⋅σ). This maximal-type characterization is an alternative to the known potential-type characterization. The subrange of the exponents q < p appeared recently in applications to nonlinear elliptic PDE with lambda _{Q} = sigma (Q) |Q|^{frac {alpha }{d}-1}, α ∈ (0, d). In this important special case Tλ is a discrete analogue of the Riesz potential I_{alpha }=(-{Delta })^{-frac {alpha }{2}}, and Mλ is the dyadic fractional maximal operator.

Two-weight norm inequalities for fractional integral operators with A_{lambda,infty} weights

In this paper, we introduce a new class of weights, the A_{lambda, infty} weights, which contains the classical A_{infty} weights. We prove a mixed A_{p,q}–A_{lambda,infty} type estimate for fractional integral operators.

Two-weight characterization for commutators of bi-parameter fractional integrals

Let Iα→ be the bi-parameter fractional integral operator on Rn1×Rn2, Iα→(f)(x)=∫Rn1×Rn2f(y1,y2)|x1−y1|n1−α1|x2−y2|n2−α2dy,0<αi<ni,i=1,2.In this paper, we give a characterization of two-weight norm inequality for the commutator of Iα→. We show that for μ,λ∈Ap,q(Rn→), ‖[b,Iα→]‖Lp(μp)→Lq(λq)≃‖b‖bmo(ν), where ν=μλ−1, and 1p−1q=α1n1=α1n2. It extends the recent one-parameter theory to the bi-parameter setting. We use the modern dyadic methods, in which the main idea is to represent continuous operators in terms of dyadic operators. Moreover, by introducing some new full and mixed bi-parameter dyadic paraproducts, we write the commutator as a finite linear combination of them.

Weighted Boundedness of Maximal Functions and Fractional Bergman Operators

The aim of this paper is to study two-weight norm inequalities for fractional maximal functions defined on the upper-half plane. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak-type inequalities. Our characterizations are in terms of Sawyer and Bekolle–Bonami-type conditions. We also obtain a $$\Phi $$ -bump characterization for these maximal functions, where $$\Phi $$ is a Orlicz function. As a consequence, we obtain two-weight norm inequalities for fractional Bergman operators. Finally, we provide some sharp weighted inequalities for the fractional maximal functions.

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$ g λ ∗ -Function

Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, \begin{align*} g_{\lambda}^{*}(f)(x) =\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda} |\nabla P_tf(y,t)|^2 \frac{dy dt}{t^{n-1}}\bigg)^{1/2}, \ \quad \lambda > 1, \end{align*} where $P_tf(y,t)=p_t*f(y)$, $p_t(y)=t^{-n}p(y/t)$ and $p(x) = (1+|x|^2)^{-(n+1)/2}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two-weight norm inequality for $g_{\lambda}^{*}$-function. We show that, $\big\| g_{\lambda}^{*}(f \sigma) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(\sigma)}$ if and only if the two-weight Muchenhoupt $A_2$ condition holds, and a testing condition holds : \begin{align*} \sup_{Q : cubes \ in \mathbb R^n} \frac{1}{\sigma(Q)} \int_{\mathbb R^n} \iint_{\widehat{Q}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda}|\nabla P_t(\mathbf{1}_Q \sigma)(y,t)|^2 \frac{w dx dt}{t^{n-1}} dy < \infty, \end{align*} where $\widehat{Q}$ is the Carleson box over $Q$ and $(w, \sigma)$ is a pair of weights. We actually prove this characterization for $g_{\lambda}^{*}$-function associated with more general fractional Poisson kernel $p^\alpha(x) = (1+|x|^2)^{-{(n+\alpha)}/{2}}$. Moreover, the corresponding results for intrinsic $g_{\lambda}^*$-function are also presented.

Two-Weight Norm, Poincaré, Sobolev and Stein–Weiss Inequalities on Morrey Spaces

We establish two-weight norm inequalities for singular integral operators and fractional integral operators on Morrey spaces. As a consequence of these inequalities, we obtain two-weight Poincaré and Sobolev inequalities on Morrey spaces. Moreover, we also establish the Stein–Weiss inequality, the Hardy inequality and the Rellich inequality on Morrey spaces.

Two-weight entropy boundedness of multilinear fractional type operators

This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_{\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\alpha}$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_\alpha$ and $\mathcal{I}_\alpha$ will be given.

Two-weight entropy boundedness of multilinear fractional type operators

This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_{\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\alpha}$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_\alpha$ and $\mathcal{I}_\alpha$ will be given.

Commutators in the two-weight setting

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to L^p(\mathbb{R}^n;\lambda)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $\mu$ and $\lambda$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $\lambda$ and $\mu$ being Lebesgue measure, and Bloom in the case of dimension one.

Two-Weight Norm Inequalities for the Local Maximal Function

For a local maximal function defined on a certain family of cubes lying “well inside” of \(\Omega \), a proper open subset of \({\mathbb {R}}^n\), we characterize the couple of weights (u, v) for which it is bounded from \(L^p(v)\) on \(L^q(u)\).

Two-Weight Norm Inequality for the One-Sided Hardy-Littlewood Maximal Operators in Variable Lebesgue Spaces

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.

Two-weight norm inequalities on Morrey spaces

A description of all the admissible weights similar to the Muckenhoupt class $A_p$ is an open problem for the weighted Morrey spaces. In this paper necessary condition and sufficient condition for two-weight norm inequalities on Morrey spaces to hold are given for the Hardy-Littlewood maximal operator. Necessary and sufficient condition is also verified for the power weights.

Logarithmic bump conditions for Calderón-Zygmund operators on spaces of homogeneous type

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 conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this setting. The other interesting feature of our approach is that we are able to prove the separated bump results (which always imply the corresponding double bump results) as a consequence of the double bump theorem.

Weighted estimates for vector-valued commutators of generalized fractional integrals

In this paper, the authors study the vector-valued commutator of generalized fractional integral operator Ik α,b,q where the kernel Kα satisfies some conditions associated with the Young functions. The authors prove the two-weight norm inequalities for Ik α,b,q where the weight ω is only a local integrable function. As an application of the main theorems in this paper, the weighted boundedness for vector-valued commutators of fractional integral with a rough kernel is also given. Mathematics subject classification (2010): 42B20, 42B25.

Two-weight norm inequalities for potential type integral operators in the case p>q>0 and p>1

Two-weight norm inequalities for potential type integral operators in the case p>q>0 and p>1

Two-weight norm inequalities for potential type and maximal operators in a metric space

We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hytonen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions into spaces of homogeneous type.