Displaying similar documents to “Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces”

Boundedness of para-product operators on spaces of homogeneous type

Yayuan Xiao (2017)

Czechoslovak Mathematical Journal

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We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H p ( 𝒳 ) for 1 / ( 1 + ϵ ) < p 1 , where 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ϵ is the regularity exponent of the kernel of the singular integral operator T . Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast...

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

Shuichi Sato (2019)

Czechoslovak Mathematical Journal

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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on n . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted L p spaces, 1 < p < , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

L p ( ) boundedness for the commutator of a homogeneous singular integral operator

Guoen Hu (2003)

Studia Mathematica

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The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that Ω L ( l o g L ) k + 1 ( S n - 1 ) is a sufficient condition for the kth order commutator to be bounded on L p ( ) for all 1 < p < ∞. The corresponding maximal operator is also considered.

Radial maximal function characterizations for Hardy spaces on RD-spaces

Loukas Grafakos, Liguang Liu, Dachun Yang (2009)

Bulletin de la Société Mathématique de France

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An RD-space 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type 𝒳 having “dimension” n , there exists a p 0 ( n / ( n + 1 ) , 1 ) such that for certain classes of distributions, the L p ( 𝒳 ) quasi-norms of their radial maximal functions and grand maximal functions are equivalent when p ( p 0 , ] . This result yields a radial maximal function characterization for Hardy spaces on 𝒳 . ...

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on d which satisfies μ(B(x,r)) ≤ Crⁿ for any x d and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with A p ϱ ( μ ) weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with A ϱ ( μ ) weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

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Let L be a non-negative self-adjoint operator acting on L 2 ( n ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A r weight on n × n , 1 < r < . In this article we obtain a weighted atomic decomposition for the weighted Hardy space H L , w p ( n × n ) , 0 < p 1 associated to L . Based on the atomic decomposition, we show the dual relationship between H L , w 1 ( n × n ) and BMO L , w ( n × n ) .

Weighted boundedness of Toeplitz type operators related to singular integral operators with non-smooth kernel

Xiaosha Zhou, Lanzhe Liu (2013)

Colloquium Mathematicae

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Some weighted sharp maximal function inequalities for the Toeplitz type operator T b = k = 1 m T k , 1 M b T k , 2 are established, where T k , 1 are a fixed singular integral operator with non-smooth kernel or ±I (the identity operator), T k , 2 are linear operators defined on the space of locally integrable functions, k = 1,..., m, and M b ( f ) = b f . The weighted boundedness of T b on Morrey spaces is obtained by using sharp maximal function inequalities.

Anisotropic classes of homogeneous pseudodifferential symbols

Árpád Bényi, Marcin Bownik (2010)

Studia Mathematica

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We define homogeneous classes of x-dependent anisotropic symbols γ , δ m ( A ) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent...

On weighted Hardy spaces on the unit disk

Evgeny A. Poletsky, Khim R. Shrestha (2015)

Banach Center Publications

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In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces H u p . We also provide a reduction of H problems to H u p problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein (2001)

Studia Mathematica

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Let M S denote the strong maximal operator. Let M x and M y denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that Q M y M x h < but Q M x M y h = . It is shown that if f is a function supported on Q such that Q M y M x f < but Q M x M y f = , then there exists a set A of finite measure in ℝ² such that A M S f = .

Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators

Qingying Xue (2013)

Studia Mathematica

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The following iterated commutators T , Π b of the maximal operator for multilinear singular integral operators and I α , Π b of the multilinear fractional integral operator are introduced and studied: T , Π b ( f ) ( x ) = s u p δ > 0 | [ b , [ b , [ b m - 1 , [ b , T δ ] ] m - 1 ] ] ( f ) ( x ) | , I α , Π b ( f ) ( x ) = [ b , [ b , [ b m - 1 , [ b , I α ] ] m - 1 ] ] ( f ) ( x ) , where T δ are the smooth truncations of the multilinear singular integral operators and I α is the multilinear fractional integral operator, b i B M O for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

Weighted H p spaces

José García-Cuerva

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CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted H p spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................

Boundedness of commutators of strongly singular convolution operators on Herz-type spaces

Zongguang Liu (2003)

Studia Mathematica

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The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, T b m , on Herz spaces K ̇ q α , p ( ) and K q α , p ( ) , and on a new class of Herz-type Hardy spaces H K ̇ q , b , m α , p , 0 ( ) and H K q , b , m α , p , 0 ( ) , where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

One-sided discrete square function

A. de la Torre, J. L. Torrea (2003)

Studia Mathematica

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Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average A f ( x ) = 2 - n x x + 2 f . The square function is defined as S f ( x ) = ( n = - | A f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 . The local version of this operator, namely the operator S f ( x ) = ( n = - 0 | A f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps L p into itself (p > 1) and L into BMO. We prove that the operator S not only maps L into BMO but it also maps BMO into BMO. We also prove that the L p boundedness...

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces B p , q s ( , w ) and F p , q s ( , w ) where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

Gladis Pradolini, Jorgelina Recchi (2018)

Czechoslovak Mathematical Journal

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Let μ be a nonnegative Borel measure on d satisfying that μ ( Q ) l ( Q ) n for every cube Q n , where l ( Q ) is the side length of the cube Q and 0 < n d . We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ . Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result...

Some weighted norm inequalities for a one-sided version of g * λ

L. de Rosa, C. Segovia (2006)

Studia Mathematica

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We study the boundedness of the one-sided operator g λ , φ between the weighted spaces L p ( M ¯ w ) and L p ( w ) for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of g λ , φ . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of g λ , φ from L p ( ( M ¯ ) [ p / 2 ] + 1 w ) to L p ( w ) , where ( M ¯ ) k denotes the operator M¯ iterated k times.

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

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Let A α f ( x ) = | B ( 0 , | x | ) | - α / n B ( 0 , | x | ) f ( t ) d t be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that A α is bounded from L p to L p α with p α = n p / ( α p - n p + n ) when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space S α , Y , which is strictly larger than X, and a ’target’ space T Y , which is strictly smaller than Y, under the assumption that A α is bounded from X into Y and the Hardy-Littlewood...

Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index p Φ ( 0 , 1 ] and ω A l o c ( ) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space h ω Φ ( ) via the local grand maximal function. Let ρ ( t ) t - 1 / Φ - 1 ( t - 1 ) for all t ∈ (0,∞). We also introduce the BMO-type space b m o ρ , ω ( ) and establish the duality between h ω Φ ( ) and b m o ρ , ω ( ) . Characterizations of h ω Φ ( ) , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

Commutators of Littlewood-Paley [...] g κ ∗ g κ * -functions on non-homogeneous metric measure spaces

Guanghui Lu, Shuangping Tao (2017)

Open Mathematics

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The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ κ , b * generated by the Littlewood-Paley operator [...] Mκ∗ κ * and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ κ * satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ κ , b * is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...

Embeddings between weighted Copson and Cesàro function spaces

Amiran Gogatishvili, Rza Mustafayev, Tuğçe Ünver (2017)

Czechoslovak Mathematical Journal

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In this paper, characterizations of the embeddings between weighted Copson function spaces Cop p 1 , q 1 ( u 1 , v 1 ) and weighted Cesàro function spaces Ces p 2 , q 2 ( u 2 , v 2 ) are given. In particular, two-sided estimates of the optimal constant c in the inequality d ( 0 0 t f ( τ ) p 2 v 2 ( τ ) d τ q 2 / p 2 u 2 ( t ) d t ) 1 / q 2 c 0 t f ( τ ) p 1 v 1 ( τ ) d τ q 1 / p 1 u 1 ( t ) d t 1 / q 1 , d where p 1 , p 2 , q 1 , q 2 ( 0 , ) , p 2 q 2 and u 1 , u 2 , v 1 , v 2 are weights on ( 0 , ) , are obtained. The most innovative part consists of the fact that possibly different parameters p 1 and p 2 and possibly different inner weights v 1 and v 2 are allowed. The proof is based on the combination of duality techniques...

Note on duality of weighted multi-parameter Triebel-Lizorkin spaces

Wei Ding, Jiao Chen, Yaoming Niu (2019)

Czechoslovak Mathematical Journal

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We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces F ˙ p α , q ( ω ; n 1 × n 2 ) . This space has been introduced and the result ( F ˙ p α , q ( ω ; n 1 × n 2 ) ) * = CMO p - α , q ' ( ω ; n 1 × n 2 ) for 0 < p 1 has been proved in Ding, Zhu (2017). In this paper, for 1 < p < , 0 < q < we establish its dual space H ˙ p α , q ( ω ; n 1 × n 2 ) .

Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions

Fabio Berra (2022)

Czechoslovak Mathematical Journal

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We give a quantitative characterization of the pairs of weights ( w , v ) for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak ( p , p ) type inequality for 1 p < . More precisely, given any measurable set E 0 , the estimate w ( { x n : M + , d ( 𝒳 E 0 ) ( x ) > t } ) C [ ( w , v ) ] A p + , d ( ) p t p v ( E 0 ) holds if and only if the pair ( w , v ) belongs to A p + , d ( ) , that is, | E | | Q | [ ( w , v ) ] A p + , d ( ) v ( E ) w ( Q ) 1 / p for every dyadic cube Q and every measurable set E Q + . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...