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Ferenc Móricz - One of the best experts on this subject based on the ideXlab platform.
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Statistical extension of classical Tauberian theorems in the case of logarithmic summability of Locally Integrable Functions on $[1,\infty)$
arXiv: Classical Analysis and ODEs, 2012Co-Authors: Ferenc Móricz, Zoltán NémethAbstract:Let $s:[1,\infty) \to \C $ be a Locally Integrable Function in Lebesgue's sense. The logarithmic (also called harmonic) mean of the Function $s$ is defined by [\tau(t) := \frac 1{\log t} \int_1^t \frac {s(x)}{x} dx, \qquad t>1,] where the logarithm is to base $e$. Besides the ordinary limit $\lim_{x\to \infty} s(x)$, we also use the notion of the so-called statistical limit of $s$ at $\infty$, in notation: $ \stlim_{x\to \infty} s(x)=\ell $, by which we mean that for every $\e>0$, [\lim_{b\to \infty} \frac 1b \Big | \Big {x\in(1,b): |s(x)-\ell| >\e \Big} \Big| = 0.] We also use the ordinary limit $\lim_{t\to\infty} \tau(t)$ as well as the statistical limit $\stlim_{t\to\infty} \tau(t)$. We will prove the following Tauberian theorem: Suppose that the real-valued Function $s$ is slowly decreasing or the complex-valued $s$ is slowly oscillating. If the statistical limit $\stlim_{t\to\infty} \tau(t) =\ell $ exists, then the ordinary limit $\lim_{x\to\infty} s(x) = \ell $ also exists.
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necessary and sufficient tauberian conditions for the logarithmic summability of Functions and sequences
arXiv: Classical Analysis and ODEs, 2012Co-Authors: Ferenc MóriczAbstract:Let $s: [1, \infty) \to \C$ be a Locally Integrable Function in Lebesgue's sense on the infinite interval $[1, \infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\in \C$ such that $$\lim_{t\to \infty} \tau(t) = A, \quad {\rm where} \quad \tau(t):= {1\over \log t} \int^t_1 {s(u) \over u} du.\leqno(*)$$ It is clear that if the ordinary limit $s(t) \to A$ exists, then the limit $\tau(t) \to A$ also exists as $t\to \infty$. We present sufficient conditions, which are also necessary in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability $(C,1)$. For example, if the Function $s$ is slowly oscillating, by which we mean that for every $\e>0$ there exist $t_0 = t_0 (\e) > 1$ and $\lambda=\lambda(\e) > 1$ such that $$|s(u) - s(t)| \le \e \quad {\rm whenever}\quad t_0 \le t < u \le t^\lambda,$$ then the converse implication holds true: the ordinary convergence $\lim_{t\to \infty} s(t) = A$ follows from (*). We also present necessary and sufficient Tauberian conditions under which the ordinary convergence of a numerical sequence $(s_k)$ follows from its logarithmic summability. Among others, we give a more transparent proof of an earlier Tauberian theorem due to Kwee [3].
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On the regular convergence of multiple series of numbers and multiple integrals of Locally Integrable Functions over $\bar{\R}^m_+$
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Ferenc MóriczAbstract:We investigate the regular convergence of the $m$-multiple series $$\sum^\infty_{j_1=0} \sum^\infty_{j_2=0}...\sum^\infty_{j_m=0} \ c_{j_1, j_2,..., j_m}\leqno(*)$$ of complex numbers, where $m\ge 2$ is a fixed integer. We prove Fubini's theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim's sense can be computed by successive summation. We introduce and investigate the regular convergence of the $m$-multiple integral $$\int^\infty_0 \int^\infty_0...\int^\infty_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m,\leqno(**)$$ where $f: \bar{\R}^m_+ \to \C$ is a Locally Integrable Function in Lebesgue's sense over the closed positive octant $\bar{\R}^m_+:= [0, \infty)^m$. Our main result is a generalized version of Fubini's theorem on successive integration formulated in Theorem 4.1 as follows. If $f\in L^1_{\loc} (\bar{\R}^m_+)$, the multiple integral (**) converges regularly, and $m=p+q$, where $m, p\in \N_+$, then the finite limit $$\lim_{v_{p+1},..., v_m \to \infty} \int^{v_1}_{u_1} \int^{v_2}_{u_2}...\int^{v_p}_{u_p} \int^{v_{p+1}}_0...\int^{v_m}_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m$$ $$=:J(u_1, v_1; u_2, v_2;...; u_p, v_p), \quad 0\le u_k\le v_k
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Strong Cesàro summability and statistical limit of fourier integrals
Analysis, 2005Co-Authors: Ferenc MóriczAbstract:AbstractIn our previous paper [2], we introduced the concept of statistical limit of a measurable Function and that of strong Cesàro summability of a Locally Integrable Function. As an application, we proved there that the Fourier integral of a Function ƒ ∈ L(i) We prove that if ƒ ∈ L(ii) We complete and simplify the proof of [2, Statement (γ) of Theorem 3], which says that if ƒ ∈ L
Rivera-ríos I.p. - One of the best experts on this subject based on the ideXlab platform.
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On pointwise and weighted estimates for commutators of Calderón-Zygmund operators
Advances in Mathematics, 2017Co-Authors: Lerner A. K, Ombrosi S., Rivera-ríos I.p.Abstract:In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b, T ]$ with a Locally Integrable Function $b$. This result is applied into two directions. If $b \in BMO$, we improve several weighted weak type bounds for $[b, T ]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b, T ]$ due to Bloom-Holmes-Lacey-Wick.MTM2012-3074
Lerner A. K - One of the best experts on this subject based on the ideXlab platform.
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On pointwise and weighted estimates for commutators of Calderón-Zygmund operators
Advances in Mathematics, 2017Co-Authors: Lerner A. K, Ombrosi S., Rivera-ríos I.p.Abstract:In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b, T ]$ with a Locally Integrable Function $b$. This result is applied into two directions. If $b \in BMO$, we improve several weighted weak type bounds for $[b, T ]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b, T ]$ due to Bloom-Holmes-Lacey-Wick.MTM2012-3074
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On pointwise and weighted estimates for commutators of Calder\'on-Zygmund operators
2017Co-Authors: Lerner A. K, Ombrosi Sheldy, Rivera-ríos, Israel P.Abstract:In recent years, it has been well understood that a Calder\'on-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a Locally Integrable Function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.Comment: V3: Lemma 5.1 is corrected. We would like to thank Irina Holmes for pointing out an error in the previous versio
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On pointwise and weighted estimates for commutators of Calderón–Zygmund operators
'Elsevier BV', 2017Co-Authors: Lerner A. K, Ombrosi, Sheldy Javier, Rivera Ríos, Israel PabloAbstract:In recent years, it has been well understood that a Calderón–Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator [b,T] with a Locally Integrable Function b. This result is applied into two directions. If b∈BMO, we improve several weighted weak type bounds for [b,T]. If b belongs to the weighted BMO, we obtain a quantitative form of the two-weighted bound for [b,T] due to Bloom–Holmes–Lacey–Wick.Fil: Lerner, Andrei K.. Bar-Ilan University; IsraelFil: Ombrosi, Sheldy Javier. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Rivera Ríos, Israel Pablo. Universidad del País Vasco; Españ
Ognjen Milatovic - One of the best experts on this subject based on the ideXlab platform.
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Two realizations of Schrödinger operators on Riemannian manifolds
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Ognjen MilatovicAbstract:Abstract We consider a Schrodinger differential expression P = Δ M + V on a complete Riemannian manifold ( M , g ) with metric g, where Δ M is the scalar Laplacian on M and V is a real-valued Locally Integrable Function on M. We study two self-adjoint realizations of P in L 2 ( M ) and show their equality. This is an extension of a result of S. Agmon.
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On m-accretive Schrödinger operators in Lp-spaces on manifolds of bounded geometry
Journal of Mathematical Analysis and Applications, 2006Co-Authors: Ognjen MilatovicAbstract:Abstract Let ( M , g ) be a manifold of bounded geometry with metric g . We consider a Schrodinger-type differential expression H = Δ M + V , where Δ M is the scalar Laplacian on M and V is a nonnegative Locally Integrable Function on M . We give a sufficient condition for H to have an m -accretive realization in the space L p ( M ) , where 1 p + ∞ . The proof uses Kato's inequality and L p -theory of elliptic operators on Riemannian manifolds.
Andreas E. Kyprianou - One of the best experts on this subject based on the ideXlab platform.
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Perpetual Integrals for Lévy Processes
Journal of Theoretical Probability, 2015Co-Authors: Leif Döring, Andreas E. KyprianouAbstract:Given a Levy process ξ, we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral ∞ 0 f (ξs)ds, where f is a positive Locally Integrable Function. If μ = E(ξ1 )∈ (0, ∞) and ξ has local times we prove the 0-1 law
