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Displaying similar documents to “On the Rademacher maximal function”

On a relation between norms of the maximal function and the square function of a martingale

Masato Kikuchi (2013)

Colloquium Mathematicae

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Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For f = ( f ) n , let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and | | M g | | X | | M f | | X , then | | S g | | X C | | S f | | X , where C is a constant independent of f and g.

Noncommutative fractional integrals

Narcisse Randrianantoanina, Lian Wu (2015)

Studia Mathematica

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Let ℳ be a hyperfinite finite von Nemann algebra and ( k ) k 1 be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ( k ) k 1 . For a finite noncommutative martingale x = ( x k ) 1 k n L ( ) adapted to ( k ) k 1 and 0 < α < 1, the fractional integral of x of order α is defined by setting I α x = k = 1 n ζ k α d x k for an appropriate sequence ( ζ k ) k 1 of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein (2001)

Studia Mathematica

Similarity:

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 = .

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, Ferenc Weisz (2016)

Czechoslovak Mathematical Journal

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The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L p ( d ) (in the case p > 1 ), but (in the case when 1 / p ( · ) is log-Hölder continuous and p - = inf { p ( x ) : x d } > 1 ) on the variable Lebesgue spaces L p ( · ) ( d ) , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type ( 1 , 1 ) . In the present note we generalize Besicovitch’s covering theorem for the so-called γ -rectangles. We introduce a general maximal operator M s γ , δ and with the help of generalized Φ -functions, the strong-...

A Note on the Burkholder-Rosenthal Inequality

Adam Osękowski (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate k = 0 d f k p C p ( k = 0 ( | d f k | ² | k - 1 ) ) 1 / 2 p + ( k = 0 | d f k | p ) 1 / p p , with C p = O ( p / l n p ) as p → ∞.

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

Similarity:

We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space L p , q , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant C p , q , r such that for any ϕ L p , q , | | ϕ | | r , C p , q , r | | ϕ | | p , q .

On the distance between ⟨X⟩ and L in the space of continuous BMO-martingales

Litan Yan, Norihiko Kazamaki (2005)

Studia Mathematica

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Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, | | X | | B M O s u p T | | E [ | X - X T | | T ] | | < , where the supremum is taken over all stopping times T. Define the critical exponent b(X) by b ( X ) = b > 0 : s u p T | | E [ e x p ( b ² ( X - X T ) ) | T ] | | < , where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by q ( X ) = E [ X | ] - E [ X | ] . We use q(X) to characterize the distance between ⟨X⟩ and the class L of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities 1 / 4 d ( q ( X ) , L ) b ( X ) 4 / d ( q ( X ) , L ) hold for every continuous BMO-martingale X. ...

The weak type inequality for the Walsh system

Ushangi Goginava (2008)

Studia Mathematica

Similarity:

The main aim of this paper is to prove that the maximal operator σ is bounded from the Hardy space H 1 / 2 to weak- L 1 / 2 and is not bounded from H 1 / 2 to L 1 / 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...

Maximal non λ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with unity. The notion of maximal non λ -subrings is introduced and studied. A ring R is called a maximal non λ -subring of a ring T if R T is not a λ -extension, and for any ring S such that R S T , S T is a λ -extension. We show that a maximal non λ -subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non λ -domain is given. A necessary condition...

Maximal non-pseudovaluation subrings of an integral domain

Rahul Kumar (2024)

Czechoslovak Mathematical Journal

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The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let R S be an extension of domains. Then R is called a maximal non-pseudovaluation subring of S if R is not a pseudovaluation subring of S , and for any ring T such that R T S , T is a pseudovaluation subring of S . We show that if S is not local, then there no such T exists between R and S . We also characterize maximal non-pseudovaluation subrings of a local integral domain.

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let ( h k ) k 0 be the Haar system on [0,1]. We show that for any vectors a k from a separable Hilbert space and any ε k [ - 1 , 1 ] , k = 0,1,2,..., we have the sharp inequality | | k = 0 n ε k a k h k | | W ( [ 0 , 1 ] ) 2 | | k = 0 n a k h k | | L ( [ 0 , 1 ] ) , n = 0,1,2,..., where W([0,1]) is the weak- L space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound | | Y | | W ( Ω ) 2 | | X | | L ( Ω ) , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

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 . ...

Certain simple maximal subfields in division rings

Mehdi Aaghabali, Mai Hoang Bien (2019)

Czechoslovak Mathematical Journal

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Let D be a division ring finite dimensional over its center F . The goal of this paper is to prove that for any positive integer n there exists a D ( n ) , the n th multiplicative derived subgroup such that F ( a ) is a maximal subfield of D . We also show that a single depth- n iterated additive commutator would generate a maximal subfield of D .

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies - 1 , 1 | | j = 1 n j f j ( ε ) | | p d μ ( ε ) p - 1 , 1 - 1 , 1 | | j = 1 n δ j Δ f j ( ε ) | | p d μ ( ε ) d μ ( δ ) , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and j j = 1 n and Δ = j = 1 n j are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by p ( X ) , we show that p ( X ) k = 1 n 1 / k for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

One-sided discrete square function

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

Studia Mathematica

Similarity:

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...

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

Similarity:

Let Y be a Banach space and let S L p be a subspace of an L p space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to S ( Y ) L p ( Y ) . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and e - t B is a positive contraction...

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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Let s be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis s differentiates the integral of f if s ∉ S, and D ̅ s f ( x ) = l i m s u p d i a m ( R ) 0 , x R s | R | - 1 R f = almost everywhere if s ∈ S. If the condition D ̅ s f ( x ) = holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a G δ (resp. a G δ σ ).

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.

Maximal non valuation domains in an integral domain

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Similarity:

Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S , and for any ring T such that R T S , T is a valuation subring of S . For a local domain S , the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim ( R , S ) and...

A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces

D. Cruz-Uribe, L. Diening, A. Fiorenza (2009)

Bollettino dell'Unione Matematica Italiana

Similarity:

We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L p ( ) whenever the exponent function p ( ) satisfies log-Hölder continuity conditions. We include the case where p ( ) assumes the value infinity. The same proof also shows that the fractional maximal operator M a , 0 < a < n , maps L p ( ) into L q ( ) , where 1 / p ( ) - 1 / q ( ) = a / n .

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, Alexander Stokolos (2012)

Fundamenta Mathematicae

Similarity:

Let U , . . . , U d be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of d and the associated collection of rectangular parallelepipeds in d with sides parallel to the axes and dimensions of the form n × × n d with ( n , . . . , n d ) Γ . The associated multiparameter geometric and ergodic maximal operators M and M Γ are defined respectively on L ¹ ( d ) and L¹(Ω) by M g ( x ) = s u p x R 1 / | R | R | g ( y ) | d y and M Γ f ( ω ) = s u p ( n , . . . , n d ) Γ 1 / n n d j = 0 n - 1 j d = 0 n d - 1 | f ( U j U d j d ω ) | . Given a Young function Φ, it is shown that M satisfies the weak type estimate ...

On the structure of non-dentable subsets of C ( ω ω k )

Pericles D. Pavlakos, Minos Petrakis (2011)

Studia Mathematica

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It is shown that there is no closed convex bounded non-dentable subset K of C ( ω ω k ) such that on subsets of K the PCP and the RNP are equivalent properties. Then applying the Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains a non-dentable subset L so that on L the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of C ( ω ω k ) .

On the regularity of the one-sided Hardy-Littlewood maximal functions

Feng Liu, Suzhen Mao (2017)

Czechoslovak Mathematical Journal

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In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators + and - . More precisely, we prove that + and - map W 1 , p ( ) W 1 , p ( ) with 1 < p < , boundedly and continuously. In addition, we show that the discrete versions M + and M - map BV ( ) BV ( ) boundedly and map l 1 ( ) BV ( ) continuously. Specially, we obtain the sharp variation inequalities of M + and M - , that is, Var ( M + ( f ) ) Var ( f ) and Var ( M - ( f ) ) Var ( f ) if f BV ( ) , where Var ( f ) is the total variation of f on and BV ( ) is the set of all functions f : satisfying Var ( f ) < .