Doubling Weights Research Articles

Stability of weighted norm inequalities

We show that while individual Riesz transforms are two-weight norm stable under biLipschitz change of variables on A_{\infty} weights, they are two-weight norm unstable under even rotational change of variables on doubling weights. More precisely, we show that individual Riesz transforms are unstable under a set of rotations having full measure, which includes rotations arbitrarily close to the identity. This provides an operator theoretic distinction between A_{\infty} weights and doubling weights. More generally, all iterated Riesz transforms of odd order are rotationally unstable on pairs of doubling weights, thus demonstrating the need for characterizations of iterated Riesz transform inequalities using testing conditions as appearing in the work of Nazarov, Treil and Volberg, and other works by subsets of the authors Alexis, Lacey, Sawyer, Shen, Uriarte-Tuero and Wick, as opposed to the typically stable ’bump’ conditions.

Toeplitz operators on Bergman spaces induced by doubling weights in the unit ball of [formula omitted]

In this paper, we study the boundedness and compactness of the Toeplitz operator Tμ:Aωp↦Aωq for 1<p≤q, where the weighted Bergman spaces Aωp are induced by (radially) doubling weights in the unit ball of Cn. The equivalence between the boundedness and compactness of Tμ:Aωp↦Aωq is also established under the assumption with 1<q<p. Our work extends the early work of Peláez, Rättyä, and Sierra to higher dimensions, as well as to a larger class of radial weights.

Sums of Generalized Weighted Composition Operators from Weighted Bergman Spaces Induced by Doubling Weights into Bloch-Type Spaces

The single generalized weighted composition operator Du,ψn on various spaces of analytic functions has been investigated for decades, i.e., Du,ψnf=u·(f(n)∘ψ), where f∈H(D). However, the study of the finite sum of generalized weighted composition operators with different orders, i.e., PU,ψkf=u0·f∘ψ+u1·f′∘ψ+⋯+uk·f(k)∘ψ, is far from complete. The boundedness, compactness and essential norm of sums of generalized weighted composition operators from weighted Bergman spaces with doubling weights into Bloch-type spaces are investigated. We show a rigidity property of PU,ψk. Specifically, the boundedness and compactness of the sum PU,ψk is equivalent to those of each Dun,ψn, 0≤n≤k.

Correction to: Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Correction to: Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Composition operators on weighted Bergman spaces induced by doubling weights

Composition operators on weighted Bergman spaces induced by doubling weights

Poincaré inequalities and Ap weights on bow-ties

A metric space X is called a bow-tie if it can be written as ▪, where ▪ and ▪ are closed subsets of X. We show that a doubling measure μ on X supports a (q,p)-Poincaré inequality on X if and only if X satisfies a quasiconvexity-type condition, μ supports a (q,p)-Poincaré inequality on both ▪ and ▪, and a variational ▪-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at the point x0. In particular, we study the bow-tie XRn consisting of the positive and negative hyperquadrants in Rn equipped with a radial doubling weight and characterize the validity of the ▪-Poincaré inequality on XRn in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin.

Stević–Sharma‐type operators between Bergman spaces induced by doubling weights

Using Khinchin's inequality, Geršgorin's theorem, and the atomic decomposition of Bergman spaces, we estimate the norm and essential norm of Stević–Sharma‐type operators between weighted Bergman spaces and and the sum of weighted differentiation composition operators with different symbols from the weighted Bergman spaces to . The estimates of those between Bergman spaces remove all the restrictions of a result of Stevic, Sharma, and Bhat. As a by‐product, we also get an interpolation theorem for Bergman spaces induced by doubling weights.

Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights

Abstract Let ω be a doubling weight and 0 &lt; p ≤ q &lt; ∞ {0&lt;p\leq q&lt;\infty} . The essential norm of Riemann–Stieltjes operator T g {T_{g}} from the weighted Bergman space A ω p {A^{p}_{\omega}} to A ω q {A^{q}_{\omega}} was investigated in the unit ball of ℂ n {\mathbb{C}^{n}} .

Weighted composition operators from weighted Bergman-Orlicz spaces with doubling weights to weighted Zygmund spaces

Weighted composition operators from weighted Bergman-Orlicz spaces with doubling weights to weighted Zygmund spaces

Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights

The boundedness of the small Hankel operator hfν(g)=Pν(fg‾), induced by an analytic symbol f and the Bergman projection Pν associated to ν, acting from the weighted Bergman space Aωp to Aνq is characterized on the full range 0<p,q<∞ of parameters when ω and ν belong to the class D of radial weights admitting certain two-sided doubling conditions. Moreover, an asymptotic formula for the operator norm of hfν is established in terms of a suitable norm of f(n) depending upon the inducing weights and parameters. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization Aηq=Aωp1⊙Aνp2, where 1<q,p1,p2<∞ such that q−1=p1−1+p2−1 and η˜1q≍ω˜1p1ν˜1p2. Here τ˜(r)=∫r1τ(t)dt/(1−r) for all 0≤r<1.

A T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, and optimal cancellation conditions, II

We extend the T1 theorem of David and Journé, and the corresponding optimal cancellation conditions of Stein, to pairs of doubling measures, completing the weighted theory begun in [20]. For example, when σ and ω are doubling measures satisfying the classical Muckenhoupt condition, and Kλ is a smooth λ-fractional CZ kernel, we show there exists a bounded operator Tλ:L2(σ)→L2(ω) associated with Kλif and only if there is a positive ‘cancellation’ constant AKλ(σ,ω) so that∫|x−x0|<N|∫ε<|x−y|<NKλ(x,y)dσ(y)|2dω(x)≤AKλ(σ,ω)∫|x0−y|<Ndσ(y),for all 0<ε<N and x0∈Rn, along with the dual inequality. These ‘cancellation’ conditions measure the classical L2 norm of integrals of the kernel over shells, and were shown in [20] to be equivalent to the associated T1 conditions,∫Q|Tλ(1Qσ)|2dω≤TTλ(σ,ω)∫Qdσ,for all cubes Q, and its dual, for doubling measures in the presence of the classical Muckenhoupt condition on a pair of measures.The cancellation conditions can be taken with respect to either cubes or balls, the latter resulting in a direct extension of Stein's characterization. More generally, this is extended to a weak form of Tb theorem.

Weighted [formula omitted] Markov factors with doubling weights on the ball

Let Lp,w,1≤p<∞, denote the weighted Lp space of functions on the unit ball Bd with a doubling weight w on Bd. The Markov factor for Lp,w of a polynomial P is defined by ‖|∇P|‖p,w‖P‖p,w, where ∇P is the gradient of P. We investigate the worst case Markov factors for Lp,w and prove that the degree of these factors is at most 2. In particular, for the Gegenbauer weight wμ(x)=(1−|x|2)μ−1/2,μ≥0, the exponent 2 is sharp. We also study the average case Markov factor for L2,w on random polynomials with independent N(0,σ2) coefficients and obtain that the upper bound of the average (expected) Markov factor is order degree to the 3/2, as compared to the degree squared worst case upper bound.

Volterra-type operators mapping weighted Dirichlet space into H^infty

The problem of describing the analytic functions g on the unit disc such that the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H^infty is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of T_g acting from a weighted Dirichlet space D^p_omega , induced by an upper doubling weight omega , to H^infty . We also characterize, in terms of neat conditions on omega , the upper doubling weights for which T_g: D^p_omega rightarrow H^infty is bounded (or compact) only if g is constant.

A description of 𝐴_{∞}-weights for VMO

We present a new characterization of Muckenhoupt A ∞ A_{\infty } -weights whose logarithm is in vanishing mean oscillation ( R ) (\mathbb {R}) in terms of vanishing Carleson measures on R + 2 \mathbb {R}_+^2 and vanishing doubling weights on R \mathbb {R} . This also gives a novel description of strongly symmetric homeomorphisms on the real line by using a geometric quantity.

Bernstein inequality on conic domains and triangles

We establish weighted Bernstein inequalities in Lp space for the doubling weight on the conic surface V0d+1={(x,t):‖x‖=t,x∈Rd,t∈[0,1]} as well as on the solid cone bounded by the conic surface and the hyperplane t=1, which becomes a triangle on the plane when d=1. While the inequalities for the derivatives in the t variable behave as expected, there are inequalities for the derivatives in the x variables that are stronger than what one may have expected. As an example, on the triangle {(x1,x2):x1≥0,x2≥0,x1+x2≤1}, the usual Bernstein inequality for the derivative ∂1 states that ‖ϕ1∂1f‖p,w≤cn‖f‖p,w with ϕ1(x1,x2)≔x1(1−x1−x2), whereas our new result gives ‖(1−x2)−1/2ϕ1∂1f‖p,w≤cn‖f‖p,w.The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.

Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Order boundedness and essential norm of generalized weighted composition operators on Bergman spaces with doubling weights

Order boundedness and essential norm of generalized weighted composition operators on Bergman spaces with doubling weights

The Generalized Volterra Integral Operator and Toeplitz Operator on Weighted Bergman Spaces

We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disc. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class membership of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $$H^2$$ .

Approximation and Localized Polynomial Frame on Double Hyperbolic and Conic Domains

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in Xu (J Funct Anal 281(12):109257, 2021) for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed-form formulas for the reproducing kernels. The main results provide construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.

A T1 theorem for general Calderón—Zygmund operators with comparable doubling weights, and optimal cancellation conditions

A T1 theorem for general Calderón—Zygmund operators with comparable doubling weights, and optimal cancellation conditions