The Experts below are selected from a list of 1482 Experts worldwide ranked by ideXlab platform
J. M. Aldaz - One of the best experts on this subject based on the ideXlab platform.
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Variations on the Boman covering lemma
Journal of Mathematical Analysis and Applications, 2018Co-Authors: J. M. AldazAbstract:Abstract We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let 1 p ∞ and let q be its Conjugate Exponent. We prove that the strong type ( q , q ) of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the Exponent p, and in turn, these generalizations entail the weak type ( q , q ) of the centered maximal operator. We show by example that it is possible for the uncentered maximal operator to be unbounded for all 1 s ∞ , while the conclusion of the lemma holds for every 1 p ∞ ; thus, the latter condition is much weaker. Also, the boundedness of the centered maximal operator entails weak versions of the lemma.
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Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function
Journal of Geometric Analysis, 2010Co-Authors: J. M. Aldaz, Leonardo Colzani, J. Pérez LázaroAbstract:We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Holder functions on proper subintervals of ℝ are Lip α (Mf)≤(1+α)−1Lip α (f), α∈(0,1]. On ℝ, the best bound for Lipschitz functions is $\operatorname{Lip} ( Mf) \le (\sqrt{2} -1)\operatorname{Lip}( f)$ . In higher dimensions, we determine the asymptotic behavior, as d→∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, l p balls for p=1,2,∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Holder functions, the operator norm of the maximal operator is uniformly bounded by 2−α/q , where q is the Conjugate Exponent of p=1,2, and as d→∞ the norms approach this bound. When p=∞, best constants are the same as when p=1.
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Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function
arXiv: Classical Analysis and ODEs, 2010Co-Authors: J. M. Aldaz, Leonardo Colzani, J. Pérez LázaroAbstract:We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and H\"older functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_\alpha ( Mf) \le (1 + \alpha)^{-1}\operatorname{Lip}_\alpha( f)$, $\alpha\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic behavior, as $d\to\infty$, of the norm of the maximal operator associated to cross-polytopes, euclidean balls and cubes, that is, $\ell_p$ balls for $p = 1, 2, \infty$. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and H\"older functions, the operator norm of the maximal operator is uniformly bounded by $2^{-\alpha/q}$, where $q$ is the Conjugate Exponent of $p=1,2$, and as $d\to\infty$ the norms approach this bound. When $p=\infty$, best constants are the same as when $p = 1$.
J. Pérez Lázaro - One of the best experts on this subject based on the ideXlab platform.
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Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function
Journal of Geometric Analysis, 2010Co-Authors: J. M. Aldaz, Leonardo Colzani, J. Pérez LázaroAbstract:We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Holder functions on proper subintervals of ℝ are Lip α (Mf)≤(1+α)−1Lip α (f), α∈(0,1]. On ℝ, the best bound for Lipschitz functions is $\operatorname{Lip} ( Mf) \le (\sqrt{2} -1)\operatorname{Lip}( f)$ . In higher dimensions, we determine the asymptotic behavior, as d→∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, l p balls for p=1,2,∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Holder functions, the operator norm of the maximal operator is uniformly bounded by 2−α/q , where q is the Conjugate Exponent of p=1,2, and as d→∞ the norms approach this bound. When p=∞, best constants are the same as when p=1.
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Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function
arXiv: Classical Analysis and ODEs, 2010Co-Authors: J. M. Aldaz, Leonardo Colzani, J. Pérez LázaroAbstract:We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and H\"older functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_\alpha ( Mf) \le (1 + \alpha)^{-1}\operatorname{Lip}_\alpha( f)$, $\alpha\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic behavior, as $d\to\infty$, of the norm of the maximal operator associated to cross-polytopes, euclidean balls and cubes, that is, $\ell_p$ balls for $p = 1, 2, \infty$. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and H\"older functions, the operator norm of the maximal operator is uniformly bounded by $2^{-\alpha/q}$, where $q$ is the Conjugate Exponent of $p=1,2$, and as $d\to\infty$ the norms approach this bound. When $p=\infty$, best constants are the same as when $p = 1$.
Yang Bi-cheng - One of the best experts on this subject based on the ideXlab platform.
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On an extension of a more accurate Hilbert-type inequality
2008Co-Authors: Yang Bi-chengAbstract:By introducing two parameters λ,α and two pairs of Conjugate Exponent,an extension of a more accurate Hilbert-type inequality with the best constant factor is given.The equivalent form is considered for the applications.
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A New Hilbert-type Integral Inequality with Some Parameters
2008Co-Authors: Yang Bi-chengAbstract:By introducing an independent parameter and two pairs of Conjugate Exponent and using the technic of real analysis to estimate the weight function,we establish a new Hilbert-type integral inequality with a best constant factor and its equivalent form.
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A New Hilbert-type Integral Inequality and its Best Extensions
Journal of Nanchang University, 2008Co-Authors: Yang Bi-chengAbstract:By using the way of weight function,a new Hilbert-type integral inequality with a mixed kernel is given and the constant factor is the best possible.The equivalent form and the best extended integral inequalities with two pairs of Conjugate Exponent are considered.
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A Reverse Hilbert-type Inequality with a Best Constant Factor
Journal of Xiamen University, 2007Co-Authors: Yang Bi-chengAbstract:Hilbert-type inequalities are important in analysis and its applications.In recent years,by improvement of the weight coefficient and introducing some parameters,a number of new results are established.By introducing two pairs of Conjugate Exponent parameters (p,q) and (r,s),and using the improved Euler-Maclaurin′s summation formula for estimating the weight coefficient,a reverse Hilbert-type inequality with a best constant factor is proved.As applications,the equivalent forms and some particular results are given.
Giovanni Franzina - One of the best experts on this subject based on the ideXlab platform.
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Non-local torsion functions and embeddings
Applicable Analysis, 2018Co-Authors: Giovanni FranzinaAbstract:AbstractGiven , we discuss the embedding of in . In particular, for we deduce its compactness on all open sets on which it is continuous. We then relate, for all q up the fractional Sobolev Conjugate Exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in in a suitable weak sense, for every open set . The proofs make use of a non-local Hardy-type inequality in , involving the fractional torsion function as a weight.
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Non-local Torsion functions and Embeddings
arXiv: Analysis of PDEs, 2018Co-Authors: Giovanni FranzinaAbstract:Given $s \in (0,1)$, we discuss the embedding of $\mathcal D^{s,p}_0(\Omega)$ in $L^q(\Omega)$. In particular, for $1\le q < p$ we deduce its compactness on all open sets $\Omega\subset \mathbb R^N$ on which it is continuous. We then relate, for all q up the fractional Sobolev Conjugate Exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $\Omega$ in a suitable weak sense, for every open set $\Omega$. The proofs make use of a non-local Hardy-type inequality in $\mathcal D^{s,p}_0(\Omega)$, involving the fractional torsion function as a weight.
Elvira Mascolo - One of the best experts on this subject based on the ideXlab platform.
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E.: Regularity under sharp anisotropic general growth conditions
2009Co-Authors: Giovanni Cupini, Paolo Marcellini, Elvira MascoloAbstract:Abstract. We prove boundedness of minimizers of energy-functionals, for in-stance of the anisotropic type (1) below, under sharp assumptions on the ex-ponents pi in terms of p ∗: the Sobolev Conjugate Exponent of p; i.e., p ∗ = np n−p
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Regularity under sharp anisotropic general growth conditions
Discrete & Continuous Dynamical Systems - B, 2009Co-Authors: Giovanni Cupini, Paolo Marcellini, Elvira MascoloAbstract:We prove boundedness of minimizers of energy-functionals, for in- stance of the anisotropic type (1) below, under sharp assumptions on the ex- ponents pi in terms of p � : the Sobolev Conjugate Exponent of p; i.e., p � = np n p , 1 p = 1 n P n i=1 1 pi . As a consequence, by mean of regularity results due to Lieber- man (21), we obtain the local Lipschitz-continuity of minimizers under sharp assumptions on the Exponents of anisotropic growth.