Dimension-Free $L^p$-Maximal Inequalities for Spherical Means in $\mathbb{Z}_{m+1}^N$

AbstractLet be a magnetic Schrödinger operator on ℝn, wheresatisfy some reverse Hölder conditions. Let be such that ϕ(x, ·) for any given x ∊ ℝn is an Orlicz function, ϕ( ·, t) ∊ A∞(ℝn) for all t ∊ (0,∞) (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index . In this article, the authors prove that second-order Riesz transforms VA-1 and are bounded from the Musielak–Orlicz–Hardy space Hµ,A(Rn), associated with A, to theMusielak–Orlicz space Lµ(Rn). Moreover, we establish the boundedness of VA-1 on . As applications, some maximal inequalities associated with A in the scale of Hµ,A(Rn) are obtained

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Remark on maximal inequalities for Bessel processes

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2019.123531 ◽

2020 ◽

Vol 482 (1) ◽

pp. 123531

Author(s):

Cloud Makasu

Keyword(s):

Bessel Processes ◽

Maximal Inequalities

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Maximal Inequalities and Empirical Central Limit Theorems

Empirical Process Techniques for Dependent Data ◽

10.1007/978-1-4612-0099-4_3 ◽

2002 ◽

pp. 137-159 ◽

Cited By ~ 8

Author(s):

Jérôme Dedecker ◽

Sana Louhichi

Keyword(s):

Central Limit ◽

Limit Theorems ◽

Central Limit Theorems ◽

Maximal Inequalities

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Exact bounds for tail probabilities of martingales with bounded differences

Lietuvos matematikos rinkinys ◽

10.15388/lmr.2009.73 ◽

2009 ◽

Vol 50 ◽

Author(s):

Dainius Dzindzalieta

Keyword(s):

Random Walks ◽

General Principle ◽

Isoperimetric Problem ◽

Maximal Inequality ◽

Tail Probabilities ◽

Bounds For Tail Probabilities ◽

Maximal Inequalities ◽

Exact Bounds ◽

Natural Classes

We consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem Bn(x) = supP\{Wn visits an interval [x,∞)\}, (1) where sup is taken over all possible Wn. We describe random walks which maximize the probability in (1). We also extend the results to super-martingales.For martingales our results can be interpreted as a maximalinequalitiesP\{max 1\leq k\leq n Mk \geq x\} \leq Bn(x).The maximal inequality is optimal since the equality is achieved by martingales related to the maximizing random walks. To prove the result we introduce a general principle – maximal inequalities for (natural classes of) martingales are equivalent to (seemingly weaker) inequalities for tail probabilities, in our caseBn(x) = supP{Mn \geq x}.Our methods are similar in spirit to a method used in [1], where a solution of an isoperimetric problem (1), for integer x is provided and to the method used in [4], where the isoperimetric problem of type (1) for conditionally symmetric bounded martingales was solved for all x ∈ R.

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