The Experts below are selected from a list of 5796 Experts worldwide ranked by ideXlab platform
J M Aldaz - One of the best experts on this subject based on the ideXlab platform.
-
the weak type 1 1 bounds for the Maximal Function associated to cubes grow to infinity with the dimension
Annals of Mathematics, 2011Co-Authors: J M AldazAbstract:Let Md be the centered Hardy-Littlewood Maximal Function associated to cubes in R d with Lebesgue measure, and let cd denote the lowest constant appearing in the weak type (1,1) inequality satised by Md. We show that cd ! 1 as d ! 1, thus answering, for the case of cubes, a longstanding
-
the weak type 1 1 bounds for the Maximal Function associated to cubes grow to infinity with the dimension
arXiv: Classical Analysis and ODEs, 2008Co-Authors: J M AldazAbstract:Let $M_d$ be the centered Hardy-Littlewood Maximal Function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case of cubes, a long standing open question of E. M. Stein and J. O. Str\"{o}mberg.
-
Functions of bounded variation the derivative of the one dimensional Maximal Function and applications to inequalities
Transactions of the American Mathematical Society, 2007Co-Authors: J M Aldaz, Perez J LazaroAbstract:We prove that if f: I ⊂ R→ R is of bounded variation, then the uncentered Maximal Function Mf is absolutely continuous, and its derivative satisfies the sharp inequality ∥DMf∥ L 1(I) < |Df|(I). This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.
-
Functions of bounded variation the derivative of the one dimensional Maximal Function and applications to inequalities
arXiv: Classical Analysis and ODEs, 2006Co-Authors: J M Aldaz, Perez J LazaroAbstract:We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered Maximal Function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain, under less regularity, versions of classical inequalities involving derivatives.
-
remarks on the hardy littlewood Maximal Function
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 1998Co-Authors: J M AldazAbstract:We answer questions of A. Carbery, M. Trinidad Menarguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood Maximal Function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).
Loukas Grafakos - One of the best experts on this subject based on the ideXlab platform.
-
Bilinear Spherical Maximal Function
arXiv: Classical Analysis and ODEs, 2017Co-Authors: J. A. Barrionevo, Loukas Grafakos, Danqing He, Petr Honzík, Lucas Fürstenau De OliveiraAbstract:We obtain boundedness for the bilinear spherical Maximal Function in a range of exponents that includes the Banach triangle and a range of $L^p$ with $p
-
the multilinear strong Maximal Function
Journal of Geometric Analysis, 2011Co-Authors: Loukas Grafakos, Carlos Perez, Rodolfo H TorresAbstract:A multivariable version of the strong Maximal Function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear Maximal Functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong Maximal Function, are also proved.
-
radial Maximal Function characterizations for hardy spaces on rd spaces
Bulletin de la Société Mathématique de France, 2009Co-Authors: Loukas Grafakos, Dachun YangAbstract:Un RD-espace X est un espace de type homogene au sens de Coifman et Weiss, possedant en outre une propriete de doublement inverse. Les auteurs prouvent que pour un espace de type homogene X de « dimension » n, il existe un po ∈ (n/(n+1), 1) tel que les quasi-normes L p (X) des fonctions radiales Maximales et grand-Maximales d'une certaine classe de distributions soient equivalentes lorsque p ∈ (p 0 , ∞]. Ce resultat fournit une caracterisation des espaces de Hardy sur X en termes de fonctions radiales Maximales.
-
Maximal Function characterizations of hardy spaces on rd spaces and their applications
Science China-mathematics, 2008Co-Authors: Loukas Grafakos, Dachun YangAbstract:Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a "dimension" n .F or α ∈ (0, ∞ )d enote byH p(X), H p d(X), and H ∗ ,p (X) the corresponding Hardy spaces on X defined by the nontangential Maximal Function, the dyadic Maximal Function and the grand Maximal Function, respectively. Using a new inhomogeneous Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X )w henp ∈ (1, ∞) and with each other when p ∈ (n/(n +1 ) ,1). An atomic characterization for H ∗ ,p (X )w ithp ∈ (n/(n +1 ),1) is also established; moreover, in the range p ∈ (n/(n +1 ), 1), it is proved that the spaceH ∗ ,p (X), the Hardy space H p (X) defined via the Littlewood-Paley Function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X )t o some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩(1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.
-
BEST CONSTANTS FOR UNCENTRED Maximal FunctionS
Bulletin of The London Mathematical Society, 1997Co-Authors: Loukas Grafakos, Stephen Montgomery-smithAbstract:We precisely evaluate the operator norm of the uncentred Hardy–Littlewood Maximal Function on L p (ℝ 1 ). Consequently, we compute the operator norm of the ‘strong’ Maximal Function on L p (ℝ n ), and we observe that the operator norm of the uncentred Hardy–Littlewood Maximal Function over balls on L p (ℝ n ) grows exponentially as n [xrarr ]∞.
José Madrid - One of the best experts on this subject based on the ideXlab platform.
-
Regularity of the centered fractional Maximal Function on radial Functions
arXiv: Classical Analysis and ODEs, 2019Co-Authors: David Beltran, José MadridAbstract:We study the regularity properties of the centered fractional Maximal Function $M_{\beta}$. More precisely, we prove that the map $f \mapsto |\nabla M_\beta f|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-\beta)$ if $f$ is radial Function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional Maximal Function. The main new idea consists in relating the centered and non-centered fractional Maximal Function at the derivative level.
-
Directional Maximal Function along the primes
arXiv: Classical Analysis and ODEs, 2019Co-Authors: Laura Cladek, Polona Durcik, Ben Krause, José MadridAbstract:We study a two-dimensional discrete directional Maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated Maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional Maximal operator along the integers by the first and third author.
-
endpoint sobolev continuity of the fractional Maximal Function in higher dimensions
arXiv: Classical Analysis and ODEs, 2019Co-Authors: David Beltran, José MadridAbstract:We establish continuity mapping properties of the non-centered fractional Maximal operator $M_{\beta}$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-\beta)$ the map $f \mapsto |\nabla M_\beta f|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < \beta < 1$ if $f$ is radial and for $1 \leq \beta < d$ for general $f$. The results for $1\leq \beta < d$ extend to the centered counterpart $M_\beta^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_\beta^c$ implies its continuity.
-
the variation of the fractional Maximal Function of a radial Function
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Hannes Luiro, José MadridAbstract:In this paper we study the regularity of the non-centered fractional Maximal operator $M_{\beta}$. As the main result, we prove that there exists $C(n,\beta)$ such that if $q=n/(n-\beta)$ and $f$ is a radial Function, then $\|DM_{\beta}f\|_{L^{q}(\mathbb{R}^n)}\leq C(n,\beta)\|Df\|_{L^{1}(\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $\beta=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}(\mathbb{R}^n)$.
-
Sharp inequalities for the variation of the discrete Maximal Function
Bulletin of The Australian Mathematical Society, 2016Co-Authors: José MadridAbstract:In this paper we establish new optimal bounds for the derivative of some discrete Maximal Functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for Maximal Functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].
Jean Bourgain - One of the best experts on this subject based on the ideXlab platform.
-
a note on the schrodinger Maximal Function
Journal D Analyse Mathematique, 2016Co-Authors: Jean BourgainAbstract:It is shown that control of the Schrodinger Maximal Function sup0
-
a note on the schrodinger Maximal Function
arXiv: Number Theory, 2016Co-Authors: Jean BourgainAbstract:It is shown that control of the Schrodinger Maximal Functions $\sup_{0
-
A note on the Schrodinger Maximal Function
arXiv: Number Theory, 2016Co-Authors: Jean BourgainAbstract:It is shown that control of the Schrodinger Maximal Functions $\sup_{0
-
on the schrodinger Maximal Function in higher dimension
Proceedings of the Steklov Institute of Mathematics, 2013Co-Authors: Jean BourgainAbstract:New estimates on the Maximal Function associated to the linear Schrodinger equation are established. It is shown that the almost everywhere convergence property of e itΔ f for t → 0 holds for f ∈ H s (ℝ n ), \(s > \tfrac{1} {2} - \tfrac{1} {{4n}}\), which is a new result for n ≥ 3. We also construct examples showing that \(s \geqslant \tfrac{1} {2} - \tfrac{1} {n}\) is certainly necessary when n ≥ 4. This is a further contribution to our understanding of how L. Carleson’s result for n = 1 generalizes in higher dimension. From the methodological point of view, crucial use is made of J. Bourgain and L. Guth’s results and techniques that are based on the multi-linear oscillatory integral theory developed by J. Bennett, T. Carbery and T. Tao.
-
on the hardy littlewood Maximal Function for the cube
arXiv: Functional Analysis, 2012Co-Authors: Jean BourgainAbstract:It is shown that the Hardy-Littlewood Maximal Function associated to the cube in $\mathbb R^n$ obeys dimensional free bounds in $L^p$ fir $p>1$. Earlier work only covered the range $p>\frac 32$.
Dachun Yang - One of the best experts on this subject based on the ideXlab platform.
-
Maximal Function characterizations of musielak orlicz hardy spaces associated with magnetic schrodinger operators
Frontiers of Mathematics in China, 2015Co-Authors: Dachun Yang, Dongyong YangAbstract:Let φ be a growth Function, and let A:= −(∇−ia)·(∇−ia)+V be a magnetic Schrodinger operator on L2(ℝn), n ⩾ 2, where a:= (a1, a2, …, an) ∈ Lloc2(ℝn,ℝn) and 0 ⩽ V ∈ Lloc1(ℝn). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA, φ(ℝn), defined by the Lusin area Function associated with \(\{ e^{ - t^2 A} \} _{t > 0} \), in terms of the Lusin area Function associated with \(\{ e^{ - t\sqrt A } \} _{t > 0} \), the radial Maximal Functions and the non-tangential Maximal Functions associated with \(\{ e^{ - t^2 A} \} _{t > 0} \) and \(\{ e^{ - t\sqrt A } \} _{t > 0} \), respectively. The boundedness of the Riesz transforms LkA−1/2, k ∈ {1, 2, …, n}, from HA, φ(ℝn) to Hφ(ℝn) is also presented, where Lk is the closure of \(\frac{\partial } {{\partial x_k }} \) - iak in L2(ℝn). These results are new even when φ(x, t):= ω(x)tp for all x ∈ ℝn and t ∈ (0,+∞) with p ∈ (0, 1] and ω ∈ A∞(ℝn) (the class of Muckenhoupt weights on ℝn).
-
Radial Maximal Function characterizations of hardy spaces on RD-spaces and their applications
Mathematische Annalen, 2009Co-Authors: Dachun Yang, Yuan ZhouAbstract:Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a "dimension" n. For α (0, ∞) denote by H α p (X), H d p (X), and H ,p * (X) the corresponding Hardy spaces on X defined by the nontangential Maximal Function, the dyadic Maximal Function and the grand Maximal Function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p (1,∞] and with each other when p (n/(n + 1), 1] . An atomic characterization for H *,p (X) with p (n/(n + 1), 1] is also established; moreover, in the range p (n/(n + 1),1] , it is proved that the space H ,p * (X), the Hardy space H p (X) defined via the Littlewood-Paley Function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B. © 2008 Science in China Press and Springer-Verlag GmbH.
-
radial Maximal Function characterizations for hardy spaces on rd spaces
Bulletin de la Société Mathématique de France, 2009Co-Authors: Loukas Grafakos, Dachun YangAbstract:Un RD-espace X est un espace de type homogene au sens de Coifman et Weiss, possedant en outre une propriete de doublement inverse. Les auteurs prouvent que pour un espace de type homogene X de « dimension » n, il existe un po ∈ (n/(n+1), 1) tel que les quasi-normes L p (X) des fonctions radiales Maximales et grand-Maximales d'une certaine classe de distributions soient equivalentes lorsque p ∈ (p 0 , ∞]. Ce resultat fournit une caracterisation des espaces de Hardy sur X en termes de fonctions radiales Maximales.
-
Maximal Function characterizations of hardy spaces on rd spaces and their applications
Science China-mathematics, 2008Co-Authors: Loukas Grafakos, Dachun YangAbstract:Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a "dimension" n .F or α ∈ (0, ∞ )d enote byH p(X), H p d(X), and H ∗ ,p (X) the corresponding Hardy spaces on X defined by the nontangential Maximal Function, the dyadic Maximal Function and the grand Maximal Function, respectively. Using a new inhomogeneous Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X )w henp ∈ (1, ∞) and with each other when p ∈ (n/(n +1 ) ,1). An atomic characterization for H ∗ ,p (X )w ithp ∈ (n/(n +1 ),1) is also established; moreover, in the range p ∈ (n/(n +1 ), 1), it is proved that the spaceH ∗ ,p (X), the Hardy space H p (X) defined via the Littlewood-Paley Function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X )t o some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩(1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.
