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Hilbert transform, Toeplitz operators and Hankel operators and invariant A∞ weights.

Sergei Treil, Alexander Volberg, Dechao Zheng (1997)

Revista Matemática Iberoamericana

In this paper, several sufficient conditions for boundedness of the Hilbert transform between two weighted Lp-spaces are obtained. Invariant A∞ weights are obtained. Several characterizations of invariant A∞ weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators of Hankel operators to be bounded on the Hardy space of the unit circle using Orlicz spaces and Lorentz spaces.

Images of some functions and functional spaces under the Dunkl-Hermite semigroup

Néjib Ben Salem, Walid Nefzi (2013)

Commentationes Mathematicae Universitatis Carolinae

We propose the study of some questions related to the Dunkl-Hermite semigroup. Essentially, we characterize the images of the Dunkl-Hermite-Sobolev space, 𝒮 ( ) and L α p ( ) , 1 < p < , under the Dunkl-Hermite semigroup. Also, we consider the image of the space of tempered distributions and we give Paley-Wiener type theorems for the transforms given by the Dunkl-Hermite semigroup.

Inégalités à poids pour l'opérateur de Hardy-Littlewood-Sobolev dans les espaces métriques mesurés à deux demi-dimensions

David Mascré (2006)

Colloquium Mathematicae

On a metric measure space (X,ϱ,μ), consider the weight functions w α ( x ) = ϱ ( x , z ) - α if ϱ(x,z₀) < 1, w α ( x ) = ϱ ( x , z ) - α if ϱ(x,z₀) ≥ 1, w β ( x ) = ϱ ( x , z ) - β if ϱ(x,z₀) < 1, w β ( x ) = ϱ ( x , z ) - β if ϱ(x,z₀) ≥ 1, where z₀ is a given point of X, and let κ a : X × X be an operator kernel satisfying κ a ( x , y ) c ϱ ( x , y ) a - d for all x,y ∈ X such that ϱ(x,y) < 1, κ a ( x , y ) c ϱ ( x , y ) a - D for all x,y ∈ X such that ϱ(x,y)≥ 1, where 0 < a < min(d,D), and d and D are respectively the local and global volume growth rate of the space X. We determine conditions on a, α₀, α₁, β₀, β₁ ∈ ℝ for the Hardy-Littlewood-Sobolev operator...

Integral operators and weighted amalgams

C. Carton-Lebrun, H. Heinig, S. Hofmann (1994)

Studia Mathematica

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from q ̅ ( L v p ̅ ) into q ( L u p ) . For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted L p -spaces. Amalgams of the form q ( L w p ) , 1 < p,q < ∞ , q ≠ p, w A p , are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

Intrinsic characterizations of distribution spaces on domains

V. Rychkov (1998)

Studia Mathematica

We give characterizations of Besov and Triebel-Lizorkin spaces B p q s ( ) and F p q s ( ) in smooth domains n via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s ∈ ℝ, 0 < p,q ≤ ∞ are stated in terms of the mixed norm of a certain maximal function of a distribution. For s ∈ ℝ, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ characterizations without use of maximal functions are also obtained.

John-Nirenberg lemmas for a doubling measure

Daniel Aalto, Lauri Berkovits, Outi Elina Kansanen, Hong Yue (2011)

Studia Mathematica

We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderón-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

JOP's counting function and Jones' square function

Karin Reinhold (2006)

Studia Mathematica

We study a class of square functions in a general framework with applications to a variety of situations: samples along subsequences, averages of d actions and of positive L¹ contractions. We also study the relationship between a counting function first introduced by Jamison, Orey and Pruitt, in a variety of situations, and the corresponding ergodic averages. We show that the maximal counting function is not dominated by the square functions.

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