scholarly journals On some maximal inequalities for demimartingales and N-demimartingales based on concave Young functions

2012 ◽  
Vol 396 (2) ◽  
pp. 434-440 ◽  
Author(s):  
Xuejun Wang ◽  
B.L.S. Prakasa Rao ◽  
Shuhe Hu ◽  
Wenzhi Yang
Keyword(s):  
Maximal Inequalities ◽  
Young Functions

Two-weight weak-type maximal inequalities for martingales

2009 ◽  
Vol 29 (2) ◽  
pp. 402-408 ◽  
Author(s):  
Ren Yanbo ◽  
Hou Youliang
Keyword(s):  
Weak Type ◽  
Maximal Inequalities

Maximal Inequalities for Additive Processes

2011 ◽  
Vol 25 (4) ◽  
pp. 981-1012
Author(s):  
Michael J. Klass ◽  
Ming Yang
Keyword(s):  
Maximal Inequalities ◽  
Additive Processes

Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schr ödinger Operators

2015 ◽  
Vol 58 (2) ◽  
pp. 432-448 ◽  
Author(s):  
Dachun Yang ◽  
Sibei Yang
Keyword(s):  
Hardy Space ◽  
Orlicz Space ◽  
Orlicz Function ◽  
Second Order ◽  
Riesz Transforms ◽  
Muckenhoupt Weights ◽  
Schrodinger Operator ◽  
Magnetic Schrödinger Operator ◽  
Maximal Inequalities ◽  
Reverse Hölder

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

scholarly journals Weighted Hardy operators in local generalized Orlicz-Morrey spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 522-533
Author(s):  
C. Aykol ◽  
Z.O. Azizova ◽  
J.J. Hasanov
Keyword(s):  
Morrey Space ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Strong Type ◽  
Young Functions ◽  
Weighted Hardy Operators ◽  
Hardy Operators

In this paper, we find sufficient conditions on general Young functions $(\Phi, \Psi)$ and the functions $(\varphi_1,\varphi_2)$ ensuring that the weighted Hardy operators $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ are of strong type from a local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ into another local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$. We also obtain the boundedness of the commutators of $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ from $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ to $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$.

scholarly journals Exact bounds for tail probabilities of martingales with bounded differences

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.

scholarly journals Lower bounds for Orlicz eigenvalues

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ariel Salort
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Dirichlet Boundary Conditions ◽  
Young Functions ◽  
New Strategies ◽  
Nonlinear Eigenvalue

<p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

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