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Displaying similar documents to “Variation for the Riesz transform and uniform rectifiability”

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

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We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

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Let ( X , d , μ ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ . Let L be a non-negative self-adjoint operator of order m on L 2 ( X ) . Assume that the semigroup e - t L generated by L satisfies the Davies-Gaffney estimate of order m and L satisfies the Plancherel type estimate. Let H L p ( X ) be the Hardy space associated with L . We show the boundedness of Stein’s square function 𝒢 δ ( L ) arising from Bochner-Riesz means associated to L from Hardy spaces H L p ( X ) to L p ( X ) , and also study...

The σ -property in C ( X )

Anthony W. Hager (2016)

Commentationes Mathematicae Universitatis Carolinae

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The σ -property of a Riesz space (real vector lattice) B is: For each sequence { b n } of positive elements of B , there is a sequence { λ n } of positive reals, and b B , with λ n b n b for each n . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “ σ ” obtains for a Riesz space of continuous real-valued functions C ( X ) . A basic result is: For discrete X , C ( X ) has σ iff the cardinal | X | < 𝔟 , Rothberger’s bounding number. Consequences...

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang (2016)

Czechoslovak Mathematical Journal

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Let 1 = - Δ + V be a Schrödinger operator and let 2 = ( - Δ ) 2 + V 2 be a Schrödinger type operator on n ( n 5 ) , where V 0 is a nonnegative potential belonging to certain reverse Hölder class B s for s n / 2 . The Hardy type space H 2 1 is defined in terms of the maximal function with respect to the semigroup { e - t 2 } and it is identical to the Hardy space H 1 1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator b = b f - ( b f ) generated by the Riesz transform = 2 2 - 1 / 2 , where b BMO θ ( ρ ) , which is larger...

Sharp inequalities for Riesz transforms

Adam Osękowski (2014)

Studia Mathematica

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We establish the following sharp local estimate for the family R j j = 1 d of Riesz transforms on d . For any Borel subset A of d and any function f : d , A | R j f ( x ) | d x C p | | f | | L p ( d ) | A | 1 / q , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, C p = [ 2 q + 2 Γ ( q + 1 ) / π q + 1 k = 0 ( - 1 ) k / ( 2 k + 1 ) q + 1 ] 1 / q , 1 < p < 2, and C p = [ 4 Γ ( q + 1 ) / π q k = 0 1 / ( 2 k + 1 ) q ] 1 / q , 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

L¹ representation of Riesz spaces

Bahri Turan (2006)

Studia Mathematica

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Let E be a Riesz space. By defining the spaces L ¹ E and L E of E, we prove that the center Z ( L ¹ E ) of L ¹ E is L E and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality L ¹ E = Z ( E ) ' . Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in L ¹ E which are different from the representations appearing in the literature.

A note on the commutator of two operators on a locally convex space

Edvard Kramar (2016)

Commentationes Mathematicae Universitatis Carolinae

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Denote by C the commutator A B - B A of two bounded operators A and B acting on a locally convex topological vector space. If A C - C A = 0 , we show that C is a quasinilpotent operator and we prove that if A C - C A is a compact operator, then C is a Riesz operator.

Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Hongbin Wang (2016)

Czechoslovak Mathematical Journal

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Let Ω L s ( S n - 1 ) for s 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μ Ω and b is defined by [ b , μ Ω ] ( f ) ( x ) = ( 0 | x - y | t Ω ( x - y ) | x - y | n - 1 [ b ( x ) - b ( y ) ] f ( y ) d y | 2 d t t 3 1 / 2 . In this paper, the author proves the ( L p ( · ) ( n ) , L p ( · ) ( n ) ) -boundedness of the Marcinkiewicz integral operator μ Ω and its commutator [ b , μ Ω ] when p ( · ) satisfies some conditions. Moreover, the author obtains the corresponding result about μ Ω and [ b , μ Ω ] on Herz spaces with variable exponent.

Radon-Nikodym property

Surjit Singh Khurana (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a Banach space E and a probability space ( X , 𝒜 , λ ) , a new proof is given that a measure μ : 𝒜 E , with μ λ , has RN derivative with respect to λ iff there is a compact or a weakly compact C E such that | μ | C : 𝒜 [ 0 , ] is a finite valued countably additive measure. Here we define | μ | C ( A ) = sup { k | μ ( A k ) , f k | } where { A k } is a finite disjoint collection of elements from 𝒜 , each contained in A , and { f k } E ' satisfies sup k | f k ( C ) | 1 . Then the result is extended to the case when E is a Frechet space.

Some properties of algebras of real-valued measurable functions

Ali Akbar Estaji, Ahmad Mahmoudi Darghadam (2023)

Archivum Mathematicum

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Let M ( X , 𝒜 ) ( M * ( X , 𝒜 ) ) be the f -ring of all (bounded) real-measurable functions on a T -measurable space ( X , 𝒜 ) , let M K ( X , 𝒜 ) be the family of all f M ( X , 𝒜 ) such that coz ( f ) is compact, and let M ( X , 𝒜 ) be all f M ( X , 𝒜 ) that { x X : | f ( x ) | 1 n } is compact for any n . We introduce realcompact subrings of M ( X , 𝒜 ) , we show that M * ( X , 𝒜 ) is a realcompact subring of M ( X , 𝒜 ) , and also M ( X , 𝒜 ) is a realcompact if and only if ( X , 𝒜 ) is a compact measurable space. For every nonzero real Riesz map ϕ : M ( X , 𝒜 ) , we prove that there is an element x 0 X such that ϕ ( f ) = f ( x 0 ) for every f M ( X , 𝒜 ) if ( X , 𝒜 ) is a compact measurable space....

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ( Y ) ), and ϕ a linear isometry from M into C(Y) (resp., C ( Y ) ). We show, under the assumption that Π N Π T , where Π N is...

L p -improving properties of certain singular measures on the Heisenberg group

Pablo Rocha (2022)

Mathematica Bohemica

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Let μ A be the singular measure on the Heisenberg group n supported on the graph of the quadratic function ϕ ( y ) = y t A y , where A is a 2 n × 2 n real symmetric matrix. If det ( 2 A ± J ) 0 , we prove that the operator of convolution by μ A on the right is bounded from L ( 2 n + 2 ) ( 2 n + 1 ) ( n ) to L 2 n + 2 ( n ) . We also study the type set of the measures d ν γ ( y , s ) = η ( y ) | y | - γ d μ A ( y , s ) , for 0 γ < 2 n , where η is a cut-off function around the origin on 2 n . Moreover, for γ = 0 we characterize the type set of ν 0 .