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Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms

Adam Osękowski — 2014

Open Mathematics

We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures...

On the UMD constant of the space N

Adam Osękowski — 2016

Colloquium Mathematicae

Let N ≥ 2 be a given integer. Suppose that d f = ( d f ) n 0 is a martingale difference sequence with values in N and let ( ε ) n 0 be a deterministic sequence of signs. The paper contains the proof of the estimate ( s u p n 0 | | k = 0 n ε k d f k | | N 1 ) ( l n N + l n ( 3 l n N ) ) / ( 1 - ( 2 l n N ) - 1 ) s u p n 0 | | k = 0 n d f k | | N . It is shown that this result is asymptotically sharp in the sense that the least constant C N in the above estimate satisfies l i m N C N / l n N = 1 . The novelty in the proof is the explicit verification of the ζ-convexity of the space N .

Sharp moment inequalities for differentially subordinated martingales

Adam Osękowski — 2010

Studia Mathematica

We determine the optimal constants C p , q in the moment inequalities | | g | | p C p , q | | f | | q , 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

Sharp inequalities for Riesz transforms

Adam Osękowski — 2014

Studia Mathematica

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.

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

Adam Osękowski — 2014

Bulletin of the Polish Academy of Sciences. Mathematics

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.

A sharp maximal inequality for continuous martingales and their differential subordinates

Adam Osękowski — 2013

Czechoslovak Mathematical Journal

Assume that X , Y are continuous-path martingales taking values in ν , ν 1 , such that Y is differentially subordinate to X . The paper contains the proof of the maximal inequality sup t 0 | Y t | 1 2 sup t 0 | X t | 1 . The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski — 2014

Banach Center Publications

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 .

Two Inequalities for the First Moments of a Martingale, its Square Function and its Maximal Function

Adam Osękowski — 2005

Bulletin of the Polish Academy of Sciences. Mathematics

Given a Hilbert space valued martingale (Mₙ), let (M*ₙ) and (Sₙ(M)) denote its maximal function and square function, respectively. We prove that 𝔼|Mₙ| ≤ 2𝔼 Sₙ(M), n=0,1,2,..., 𝔼 M*ₙ ≤ 𝔼 |Mₙ| + 2𝔼 Sₙ(M), n=0,1,2,.... The first inequality is sharp, and it is strict in all nontrivial cases.

Sharp Norm Inequalities for Martingales and their Differential Subordinates

Adam Osękowski — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

Suppose f = (fₙ), g = (gₙ) are martingales with respect to the same filtration, satisfying | f - f n - 1 | | g - g n - 1 | , n = 1,2,..., with probability 1. Under some assumptions on f₀, g₀ and an additional condition that one of the processes is nonnegative, some sharp inequalities between the pth norms of f and g, 0 < p < ∞, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.

Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

Adam Osękowski — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

Let f be a conditionally symmetric martingale and let S(f) denote its square function. (i) For p,q > 0, we determine the best constants C p , q such that s u p n ( | f | p ) / ( 1 + S ² ( f ) ) q C p , q . Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants C N , q ' such that s u p n ( f 2 N - 1 ) ( 1 + S ² ( f ) ) q C N , q ' . These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if the conditional...

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate Y W ( 2 ( α + 1 ) ² ) / ( 2 α + 1 ) X L . Here W is the weak- L space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

Adam Osękowski — 2013

Bulletin of the Polish Academy of Sciences. Mathematics

Assume that u, v are conjugate harmonic functions on the unit disc of ℂ, normalized so that u(0) = v(0) = 0. Let u*, |v|* stand for the one- and two-sided Brownian maxima of u and v, respectively. The paper contains the proof of the sharp weak-type estimate ℙ(|v|* ≥ 1)≤ (1 + 1/3² + 1/5² + 1/7² + ...)/(1 - 1/3² + 1/5² - 1/7² + ...) 𝔼u*. Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined...

Moment Inequality for the Martingale Square Function

Adam Osękowski — 2013

Bulletin of the Polish Academy of Sciences. Mathematics

Consider the sequence ( C ) n 1 of positive numbers defined by C₁ = 1 and C n + 1 = 1 + C ² / 4 , n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

A Note on the Burkholder-Rosenthal Inequality

Adam Osękowski — 2012

Bulletin of the Polish Academy of Sciences. Mathematics

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 → ∞.

Sharp Logarithmic Inequalities for Two Hardy-type Operators

Adam Osękowski — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: S f ( x ) = 1 / | B ( 0 , | x | ) | B ( 0 , | x | ) f ( t ) d t , T f ( x ) = 1 / | B ( x , | x | ) | B ( x , | x | ) f ( t ) d t for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

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