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Rodolfo H. Torres - One of the best experts on this subject based on the ideXlab platform.
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end point estimates for iterated commutators of Multilinear singular integrals
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Carlos Perez, Rodolfo H. Torres, Gladis Pradolini, Rodrigo TrujillogonzalezAbstract:Iterated commutators of Multilinear Calderon-Zygmund operators and pointwise multiplication with functions in $BMO$ are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted results involving the vectors weights of the Multilinear Calderon-Zygmund theory recently introduced in the literature. Some better than expected estimates for certain Multilinear operators are presented too.
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new maximal functions and multiple weights for the Multilinear calderon zygmund theory
Advances in Mathematics, 2009Co-Authors: Andrei K Lerner, Carlos Perez, Rodolfo H. Torres, Sheldy Ombrosi, Rodrigo TrujillogonzalezAbstract:A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy–Littlewood maximal function is studied. The operator is used to obtain a precise control on Multilinear singular integral operators of Calderon–Zygmund type and to build a theory of weights adapted to the Multilinear setting. A natural variant of the operator which is useful to control certain commutators of Multilinear Calderon–Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new Multilinear ones are also obtained for the commutators.
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On Multilinear singular integrals of Calderón-Zygmund type
Publicacions Matematiques, 2002Co-Authors: Loukas Grafakos, Rodolfo H. TorresAbstract:A variety of results regarding Multilinear singular Calderon-Zygmund integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new Multilinear endpoint weak type estimates, Multilinear interpolation, appropriate discrete decompositions, a Multilinear version of Schur's test, and a Multilinear version of the T1 Theorem suitable for the study of Multilinear pseudodifferential and translation invariant operators. A maximal operator associated with Multilinear singular integrals is also introduced and employed to obtain wiegted norm inequalities. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
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Multilinear calderon zygmund theory
Advances in Mathematics, 2002Co-Authors: Loukas Grafakos, Rodolfo H. TorresAbstract:Abstract A systematic treatment of Multilinear Calderon–Zygmund operators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, a Multilinear T 1 theorem, and a variety of results regarding Multilinear multiplier operators.
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sharp maximal function estimates for Multilinear singular integrals
2002Co-Authors: Carlos Perez, Rodolfo H. TorresAbstract:A new proof of a weighted norm inequality for Multilinear singu- lar integrals of Calderon-Zygmund type is presented through a more general estimate involving a sharp maximal function. An application is given to the study of certain Multilinear commutators. The study of Multilinear singular integrals of Calderon-Zygmun type continues to attract many researchers. Many results obtained parallel the linear theory of classical Calderon-Zygmund operators but new interesting phenomena have also been observed. A systematic analysis of many basic properties of such operators can be found in the article by Grafakos and Torres (GT1). See also the work of Kenig and Stein (KS) and the survey article (GT2) for further references and details. One aspect of the theory that still is being developed is the one related to the study of maximal operators associated to Multilinear singular integrals and appro- priate versions of Multilinear weighted norm inequalities. In a recent work Grafakos and Torres (GT3) have obtained Multilinear weighted norm inequalities based on a version of Cotlar's inequality in the Multilinear setting. Their approach provides Multilinear analogous of the works by Coifman (C) and Coifman and Feerman (CF). Here we present a dierent approach based on the use of a modified version of the sharp maximal function of Feerman
Anastasios N. Venetsanopoulos - One of the best experts on this subject based on the ideXlab platform.
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Uncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear Subspace Learning
IEEE transactions on neural networks, 2009Co-Authors: Konstantinos N. Plataniotis, Anastasios N. VenetsanopoulosAbstract:This paper proposes an uncorrelated Multilinear principal component analysis (UMPCA) algorithm for unsupervised subspace learning of tensorial data. It should be viewed as a Multilinear extension of the classical principal component analysis (PCA) framework. Through successive variance maximization, UMPCA seeks a tensor-to-vector projection (TVP) that captures most of the variation in the original tensorial input while producing uncorrelated features. The solution consists of sequential iterative steps based on the alternating projection method. In addition to deriving the UMPCA framework, this work offers a way to systematically determine the maximum number of uncorrelated Multilinear features that can be extracted by the method. UMPCA is compared against the baseline PCA solution and its five state-of-the-art Multilinear extensions, namely two-dimensional PCA (2DPCA), concurrent subspaces analysis (CSA), tensor rank-one decomposition (TROD), generalized PCA (GPCA), and Multilinear PCA (MPCA), on the tasks of unsupervised face and gait recognition. Experimental results included in this paper suggest that UMPCA is particularly effective in determining the low-dimensional projection space needed in such recognition tasks.
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Uncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear
2009Co-Authors: Anastasios N. VenetsanopoulosAbstract:This paper proposes an uncorrelated Multilinear principal component analysis (UMPCA) algorithm for unsuper- vised subspace learning of tensorial data. It should be viewed as a Multilinear extension of the classical principal component analysis (PCA) framework. Through successive variance max- imization, UMPCA seeks a tensor-to-vector projection (TVP) that captures most of the variation in the original tensorial input while producing uncorrelated features. The solution consists of sequential iterative steps based on the alternating projection method. In addition to deriving the UMPCA framework, this work offers a way to systematically determine the maximum number of uncorrelated Multilinear features that can be extracted by the method. UMPCA is compared against the baseline PCA solution and its five state-of-the-art Multilinear extensions, namely two-dimensional PCA (2DPCA), concurrent subspaces analysis (CSA), tensor rank-one decomposition (TROD), generalized PCA (GPCA), and Multilinear PCA (MPCA), on the tasks of unsu- pervised face and gait recognition. Experimental results included in this paper suggest that UMPCA is particularly effective in determining the low-dimensional projection space needed in such recognition tasks.
Loukas Grafakos - One of the best experts on this subject based on the ideXlab platform.
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Multilinear square functions and multiple weights
MATHEMATICA SCANDINAVICA, 2019Co-Authors: Loukas Grafakos, Parasar Mohanty, Saurabh ShrivastavaAbstract:In this paper we prove weighted estimates for a class of smooth Multilinear square functions with respect to Multilinear AP weights. In particular, we establish weighted estimates for the smooth Multilinear square functions associated with disjoint cubes of equivalent side-lengths. As a consequence, for this particular class of Multilinear square functions, we provide an affirmative answer to a question raised by Benea and Bernicot (Forum Math. Sigma 4, 2016, e26) about unweighted estimates for smooth bilinear square functions
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Carleson measures associated with families of Multilinear operators
Studia Mathematica, 2012Co-Authors: Loukas Grafakos, Lucas OliveiraAbstract:Abstract. In this work we investigate the construction of Carleson measures from families of Multilinear integral operators applied to tuples of L∞ and BMO functions. We show that if the family Rt of Multilinear operators possesses cancellation in each variable, then for BMO functions b1, . . . , bm, the measure |Rt(b1, . . . , bm)(x)|dxdt/t is Carleson. However, if the family of Multilinear operators has cancellation in all variables combined this result is still valid if bj are L∞ functions, but it may fail if bj are unbounded BMO functions, as we indicate via an example. As an application of our results we obtain a Multilinear quadratic T (1) type theorem and a Multilinear version of a quadratic T (b) theorem analogous to those in Semmes [23].
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Maximal transference and summability of Multilinear Fourier series
Journal of the Australian Mathematical Society, 2006Co-Authors: Loukas Grafakos, Petr HonzíkAbstract:We obtain a maximal transference theorem that relates almost everywhere convergence of Multilinear Fourier series to boundedness of maximal Multilinear operators. We use this and other recent results on transference and Multilinear operators to deduce Lp and almost everywhere summability of certain m-linear Fourier series. We formulate conditions for the convergence of Multilinear series and we investigate certain kinds of summation.
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On Multilinear singular integrals of Calderón-Zygmund type
Publicacions Matematiques, 2002Co-Authors: Loukas Grafakos, Rodolfo H. TorresAbstract:A variety of results regarding Multilinear singular Calderon-Zygmund integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new Multilinear endpoint weak type estimates, Multilinear interpolation, appropriate discrete decompositions, a Multilinear version of Schur's test, and a Multilinear version of the T1 Theorem suitable for the study of Multilinear pseudodifferential and translation invariant operators. A maximal operator associated with Multilinear singular integrals is also introduced and employed to obtain wiegted norm inequalities. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
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Multilinear calderon zygmund theory
Advances in Mathematics, 2002Co-Authors: Loukas Grafakos, Rodolfo H. TorresAbstract:Abstract A systematic treatment of Multilinear Calderon–Zygmund operators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, a Multilinear T 1 theorem, and a variety of results regarding Multilinear multiplier operators.
Konstantinos N. Plataniotis - One of the best experts on this subject based on the ideXlab platform.
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Uncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear Subspace Learning
IEEE transactions on neural networks, 2009Co-Authors: Konstantinos N. Plataniotis, Anastasios N. VenetsanopoulosAbstract:This paper proposes an uncorrelated Multilinear principal component analysis (UMPCA) algorithm for unsupervised subspace learning of tensorial data. It should be viewed as a Multilinear extension of the classical principal component analysis (PCA) framework. Through successive variance maximization, UMPCA seeks a tensor-to-vector projection (TVP) that captures most of the variation in the original tensorial input while producing uncorrelated features. The solution consists of sequential iterative steps based on the alternating projection method. In addition to deriving the UMPCA framework, this work offers a way to systematically determine the maximum number of uncorrelated Multilinear features that can be extracted by the method. UMPCA is compared against the baseline PCA solution and its five state-of-the-art Multilinear extensions, namely two-dimensional PCA (2DPCA), concurrent subspaces analysis (CSA), tensor rank-one decomposition (TROD), generalized PCA (GPCA), and Multilinear PCA (MPCA), on the tasks of unsupervised face and gait recognition. Experimental results included in this paper suggest that UMPCA is particularly effective in determining the low-dimensional projection space needed in such recognition tasks.
Maite Fernández-unzueta - One of the best experts on this subject based on the ideXlab platform.
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Lipschitz p-summing Multilinear operators
Journal of Functional Analysis, 2020Co-Authors: Jorge Carlos Angulo-lópez, Maite Fernández-unzuetaAbstract:Abstract We apply the geometric approach provided by Σ-operators to develop a theory of p-summability for Multilinear operators. In this way, we introduce the notion of Lipschitz p-summing Multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pelczynsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type tensor norm. Moreover, we introduce the notion of Dunford-Pettis Multilinear operators. With them, we characterize when a projective tensor product contains l 1 . Relations between Lipschitz p-summing Multilinear operators with Dunford-Pettis and Hilbert-Schmidt Multilinear operators are given.
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Multilinear Operators Factoring through Hilbert Spaces
Banach Journal of Mathematical Analysis, 2019Co-Authors: Maite Fernández-unzueta, Samuel García-hernándezAbstract:We characterize those bounded Multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapie\'{n}, from the linear to the Multilinear setting. We prove that Hilbert-Schmidt and Lipschitz $2$-summing Multilinear operators naturally factor through a Hilbert space. It is also proved that the class $\Gamma$ of all Multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $\gamma$ which is in duality with $\Gamma$.
