Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 18

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last

Wyniki wyszukiwania

help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote

Sharp inequalities for Riesz transforms

100%
Osękowski A.
Studia Mathematica
|
2014
|
tom 222
|
nr 1
1-18
EN
We establish the following sharp local estimate for the family ${R_{j}}_{j=1}^{d}$ of Riesz transforms on $ℝ^{d}$. For any Borel subset A of $ℝ^{d}$ and any function $f: ℝ^{d} → ℝ$, $∫_{A} |R_{j}f(x)|dx ≤ C_{p}||f||_{L^{p}(ℝ^{d})}|A|^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = [2^{q+2}Γ(q+1)/π^{q+1} ∑_{k=0}^{∞} (-1)^{k}/(2k+1)^{q+1}]^{1/q}$, 1 < p < 2, and $C_{p}= [4Γ(q+1)/π^{q} ∑_{k=0}^{∞} 1/(2k+1)^{q}]^{1/q}$, 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.
2
Content available remote

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

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
|
tom 62
|
nr 2
187-196
EN
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 𝓗 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 𝓗-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.
3
Content available remote

A Weak-Type Inequality for Submartingales and Itô Processes

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 1
73-88
EN
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 $∥Y∥_{W} ≤ (2(α+1)²)/(2α+1) ∥X∥_{L^∞}$. 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.
4
Content available remote

Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 3
209-218
EN
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 on Euclidean domains. The proof exploits a novel estimate for orthogonal martingales satisfying differential subordination.
5
Content available remote

Sharp moment inequalities for differentially subordinated martingales

100%
Osękowski A.
Studia Mathematica
|
2010
|
tom 201
|
nr 2
103-131
EN
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 ≤ supₙ𝔼Φ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.
6
Content available remote

A Note on the Burkholder-Rosenthal Inequality

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2012
|
tom 60
|
nr 2
177-185
EN
Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $∥∑_{k=0}^{∞} df_k∥_p ≤ C_p {∥(∑_{k=0}^{∞} 𝔼 (|df_k|²| ℱ_{k-1}))^{1/2}∥_p + ∥(∑_{k=0}^{∞} |df_k|^p)^{1/p}∥_p},$ with $C_p = O(p/lnp)$ as p → ∞.
7
Content available remote

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 3
261-274
EN
Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $ℙ(sup_{t≥0} |Y_t| ≥ 1) ≤ 3.375... ∥X∥₁$. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.
8
Content available remote

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 1
81-89
EN
Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λℙ(S(f) ≥ λ) ≤ π/2 ∥f∥₁$, and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.
9
Content available remote

Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
|
tom 58
|
nr 1
65-77
EN
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 $sup_n 𝔼 (|fₙ|^p)/(1+Sₙ²(f))^q ≤ C_{p,q}$. 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 $sup_n 𝔼 (fₙ^{2N-1})(1+Sₙ²(f))^q ≤ C'_{N,q}$. 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 symmetry is not assumed.
10
Content available remote

Sharp Norm Inequalities for Martingales and their Differential Subordinates

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom 55
|
nr 4
373-385
EN
Suppose f = (fₙ), g = (gₙ) are martingales with respect to the same filtration, satisfying $|fₙ-f_{n-1}| ≤ |gₙ-g_{n-1}|$, 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.
11
Content available remote

On the UMD constant of the space $ℓ₁^{N}$

100%
Osękowski A.
Colloquium Mathematicum
|
2016
|
tom 142
|
nr 1
135-147
EN
Let N ≥ 2 be a given integer. Suppose that $df = (dfₙ)_{n≥0}$ is a martingale difference sequence with values in $ℓ₁^{N}$ and let $(εₙ)_{n≥0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $ℙ(sup_{n≥0} ||∑_{k=0}^{n} ε_{k} df_{k}||_{ℓ₁^{N}} ≥ 1) ≤ (ln N + ln(3ln N))/(1 - (2ln N)^{-1}) sup_{n≥0} 𝔼||∑_{k=0}^{n} df_{k}||_{ℓ₁^{N}}$. It is shown that this result is asymptotically sharp in the sense that the least constant $C_{N}$ in the above estimate satisfies $lim_{N→ ∞} C_{N}/ln N = 1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $ℓ₁^{N}$.
12
Content available remote

Sharp Logarithmic Inequalities for Two Hardy-type Operators

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 3
237-247
EN
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: $Sf(x) = 1/|B(0,|x|)| ∫_{B(0,|x|)} f(t)dt$, $Tf(x) = 1/|B(x,|x|)| ∫_{B(x,|x|)} f(t)dt$ 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.
13
Content available remote

Logarithmic inequalities for second-order Riesz transforms and related Fourier multipliers

100%
Osękowski A.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 1
103-126
EN
We study logarithmic estimates for a class of Fourier multipliers which arise from a nonsymmetric modulation of jumps of Lévy processes. In particular, this leads to corresponding tight bounds for second-order Riesz transforms on $ℝ^{d}$.
14
Content available remote

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

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom 53
|
nr 4
441-449
EN
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.
15
Content available remote

Weak-type inequalities for maximal operators acting on Lorentz spaces

100%
Osękowski A.
Banach Center Publications
|
2014
|
tom 101
|
nr 1
145-162
EN
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 $ϕ ∈ L^{p,q}$, $||ℳ ϕ||_{r,∞} ≤ C_{p,q,r}||ϕ||_{p,q}$.
16
Content available remote

Maximal Inequalities for Stochastic Integrals

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
|
tom 58
|
nr 3
273-287
17
Content available remote

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

100%
Osękowski A.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1198-1213
EN
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 space of matrices of dimension 2×2, and some transference-type arguments.
18
Content available remote

Moment Inequality for the Martingale Square Function

100%
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 2
169-180
EN
Consider the sequence $(Cₙ)_{n≥1}$ of positive numbers defined by C₁ = 1 and $C_{n+1} = 1 + Cₙ²/4$, n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound 𝔼 |Mₙ|≤ Cₙ𝔼 Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.