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Weak Type Inequality for the Square Function of a Nonnegative Submartingale

100%

Osękowski A.

nr 1

81-89

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.

Weak-type inequalities for maximal operators acting on Lorentz spaces

100%

Osękowski A.

nr 1

145-162

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 $ϕ ∈ L^{p,q}$, $||ℳ ϕ||_{r,∞} ≤ C_{p,q,r}||ϕ||_{p,q}$.

A Note on the Burkholder-Rosenthal Inequality

100%

Osękowski A.

tom 60

nr 2

177-185

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 → ∞.

Sharp Logarithmic Inequalities for Two Hardy-type Operators

100%

Osękowski A.

tom 63

nr 3

237-247

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.

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

100%

Osękowski A.

nr 8

1198-1213

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.

Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

100%

Osękowski A.

tom 58

nr 1

65-77

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.