Rodolfo Torres

University of California, Riverside | UCR · Office of Research / Department of Mathematics

A brief biographical account of Guido Weiss’ life; his accomplishments, mathematics, and diverse interests; and several anecdotes of his interactions with family, friends, colleagues, and students.

2019, Springer-Verlag GmbH Germany, part of Springer Nature. We present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known re...

We present the details of an extension of known results about the compactness of the commutators of bilinear Calderón–Zygmund operators with multiplication by appropriate functions in the John–Nirenberg space, to the case when the target of the operators is a quasi-Banach Lebesgue space.

Solutions to boundary value problems for Maxwell’s equations are described using the layer potential Maxwell operator. The Maxwell, electric and magnetic boundary value problems for time harmonic electromagnetic wave propagation have unique solutions in C 1 domains and satisfy optimal estimates. Sobolev-Besov regularity results hold for domains wit...

The commutators of bilinear Calder\'on-Zygmund operators and point-wise multiplication with a symbol in $cmo$ are bilinear compact operators on product of Lebesgue spaces. This work shows that, for certain non-degenerate Calder\'on-Zygmund operators, the symbol being in $cmo$ is not only sufficient but actually necessary for the compactness of the...

This work explores new deep connections between John-Nirenberg type inequalities and Muckenhoupt weight invariance for a large class of $BMO$-type spaces. The results are formulated in a very general framework in which $BMO$ spaces are constructed using a base of sets, used also to define weights with respect to a non-negative measure (not necessar...

Weighted norm inequalities for operators corresponding to non-smooth versions of Calderón-Zygmund and fractional integral multilinear operators are revisited and improved in a unified way. Graded classes of weights matching the amount of regularity assumptions on the operators are also studied.

We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.

We present a brief recount of the discrete version of Calderón’s reproducing formula as developed by M. Frazier and B. Jawerth, starting from the historical result of Calderón and leading to some of the motivation and applications of the discrete version.

We present some snapshots of Cora Sadosky’s career focusing on her intertwined roles as mathematician, mentor, and leader in the profession. We recount some of her contributions to specific areas of mathematics as well as her broader impact on the mathematical profession.

Several new results about Leibniz’s rule, sampling, and wavelets in the context of mixed Lebesgue spaces are presented. The results obtained rely in part on vector-valued estimates for Calderón-Zygmund operators and the use of appropriate maximal functions and Littlewood-Paley estimates.

The compactness of the commutators of bilinear fractional integral operators and point-wise multiplication, acting on products of Lebesgue spaces, is characterized in terms of appropriate mean oscillation properties of their symbols. The compactness of the commutators when acting on product of weighted Lebesgue spaces is also studied.

Minimal regularity conditions on the kernels of bilinear operators are identified and shown to be sufficient for the existence of end-point estimates within the context of the bilinear Calderón-Zygmund theory.

The non-pigmentary colors of the tissues of living organisms are produced by the physical interaction of light with nanostructures in the tissues. Contrary to what has been previously assumed for many decades, it has been established now that many of the beautiful blue and green colors observed in the tissues of mammals, birds, and butterflies are...

Commutators of bilinear Calder\'on-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact on appropriate products of weighted Lebesgue spaces.

Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional integral versions of the bilinear Hilbert transform yield separately compact operators.

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

Boundedness properties for pseudodifferential operators with symbols in the bilinear H\"ormander classes of sufficiently negative order are proved. The results are obtained in the scale of Lebesgue spaces and, in some cases, end-point estimates involving weak-type spaces and BMO are provided as well. From the Lebesgue space estimates, Sobolev ones...

Let T be an arbitrary operator bounded from Lp0(w) into Lp0, ∞(w) for every weight w in the Muckenhoupt class Ap0. It is proved in this article that the distribution function of Tf with respect to any weight u can be essentially majorized by the distribution function of Mf with respect to u (plus an integral term easy to control). As a consequence,...

Boundedness properties for bilinear paraproducts on several function spaces are presented. The methods are based on the realization of paraproducts as bilinear Calderón-Zygmund operators and the molecular character-ization of function spaces. This provides a unified approach for the study of paraproducts, recovering some know results and establishi...

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 lit...

The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other...

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...

This chapter contains an exposition of research done on the analysis of oscillation in signals in connection with the classical sampling theorem. Spaces of samples that quantify oscillation are described and their main properties are presented. The subject covered also relates to aspects about wavelet representation and nonlinear approximation of f...

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expan...

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of t...

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 Calderón–Zygmund type and to build a theory of weights adapted to the multilinear...

Los colores estructurales ultravioleta (UV) de las plumas de las aves son producidos por la queratina medular esponjosa de las barbas de las plumas. Varios mecanismos físicos se han propuesto para explicar la producción de estos colores, incluyendo difusión de Rayleigh, difusión de Mie y difusión coherente (i.e. interferencia constructiva). Para po...

Some recent results for bilinear or multilinear singular integrals operators are presented. The focus is on some of the results that can be viewed as natural counterparts of classical theorems in Calderon-Zygmund theory, adding to the already existing extensive literature in the subject. In particular, two different classes of operators that can be...

We prove a T(1) theorem for bilinear singular integral operators (trilinear forms) with a one-dimensional modulation symmetry.

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the biline...

The structural colours of butterflies and moths (Lepidoptera) have been attributed to a diversity of physical mechanisms, including multilayer interference, diffraction, Bragg scattering, Tyndall scattering and Rayleigh scattering. We used fibre optic spectrophotometry, transmission electron microscopy (TEM) and 2D Fourier analysis to investigate t...

For nearly 80 years, the non-iridescent, blue, integumentary structural colours of dragonflies and damselflies (Odonata) have been attributed to incoherent Tyndall or Rayleigh scattering. We investigated the production of the integumentary structural colours of a damselfly--the familiar bluet, Enallagma civile (Coenagrionidae)--and a dragonfly--the...

The boundedness of Calderón–Zygmund operators is proved in the scale of the mixed Lebesgue spaces. As a consequence, the boundedness of the bilinear null forms Qi j (u,υ) =∂i u∂j υ - ∂j u∂i υ, Q0(u,υ)=ut υt -∇x u· ∇x υ on various space–time mixed Sobolev–Lebesgue spaces is shown.

For more than a century, the blue structural colours of mammalian skin have been hypothesized to be produced by incoherent, Rayleigh or Tyndall scattering. We investigated the colour, anatomy, nanostructure and biophysics of structurally coloured skin from two species of primates - mandrill (Mandrillus sphinx) and vervet monkey (Cercopithecus aethi...

Several results and techniques that generate bilinear alternatives of a celebrated theorem of Calderon and Vaillancourt about the L2 continuity of linear pseudodierential operators with symbols with bounded derivatives are presented. The classes of bilinear pseudodierential symbols considered are shown to produce continuous operators from L2 ◊ L2 i...

We establish uniqueness in the inverse conductivity problem for conductivities which have 3/2 derivatives in L p , p > 2n. Our results are in dimensions three and higher.

Structural colours of avian skin have long been hypothesized to be produced by incoherent (Rayleigh/Tyndall) scattering. We investigated the colour, anatomy, nanostructure and biophysics of structurally coloured skin, ramphotheca and podotheca from 31 species of birds from 17 families in 10 orders from across Aves. Integumentary structural colours...

The fundamental dichotomy between incoherent (phase independent) and coherent (phase dependent) light scattering provides the best criterion for a classification of biological structural color production mechanisms. Incoherent scattering includes Rayleigh, Tyndall, and Mie scattering. Coherent scattering encompasses interference, reinforcement, thi...

The so–called special atom space Ḃ0,11 (ℝ) plays an important role as a substitute in many problems in analysis for the larger Hardy space H1(ℝ). In this article the analogous space Ḃ0,11 (ℤ) in the discrete setting is investigated. In particular, appropriate special atom decompositions of the elements in the space are obtained. The work is motivat...

A symbolic calculus for the transposes of a class of bilinear pseudodifferential operators is developed. The calculus is used to obtain boundedness results on products of Lebesgue spaces. A larger class of pseudodifferential operators that does not admit a calculus is also considered. Such a class is the bilinear analog of the so-called exotic clas...

A new proof of a weighted norm inequality for multilinear singu-lar integrals of Calderón-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.

Ultraviolet (UV) structural colors of avian feathers are produced by the spongy medullary keratin of feather barbs, but various physical mechanisms have been hypothesized to produce those colors, including Rayleigh scattering, Mie scattering, and coherent scattering (i.e. constructive interference). We used two-dimensional Fourier analysis of trans...

A variety of results regarding multilinear singular Calderón-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...

A systematic treatment of multilinear Calderón–Zygmund operators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, a multilinear T1 theorem, and a variety of results regarding multilinear multiplier operators.

A variety of results regarding multilinear Calderon-Zygmund sin- gular 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, mul- tilinear interpolation, appropriate discrete decompositions, a mul- tilinear version of Sc...

A multilinear version of Schur's test is obtained for products of Lp spaces and is used to derive boundedness for multilinear multiplier operators acting on Sobolev and Besov spaces.

. The maximal operator associated with multilinear Calder'on-Zygmund singular integrals is introduced and shown to be bounded on product of Lebesgue spaces. Moreover weighted norm inequalities are obtained for this maximal operator as well as for the corresponding singular integrals. 1. Introduction The analysis of multilinear singular integrals ha...

We investigated the anatomy, nanostructure and biophysics of the structurally coloured facial caruncles of three species in a clade of birds endemic to Madagascar (Philepittinae, Eurylaimidae: Aves). Caruncle tissues of all species had reflectance spectra with prominent, peak hues between 403 and 528 nm. Dark blue Neodrepanis tissues had substantia...

. Using discrete decomposition techniques, bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct an...

We conducted two-dimensional (2D) discrete Fourier analyses of the spatial variation in refractive index of the spongy medullary keratin from four different colours of structurally coloured feather barbs from three species of bird: the rose-faced lovebird, Agapornis roseicollis (Psittacidae), the budgerigar, Melopsittacus undulatus (Psittacidae), a...

The structural colours of avian feather barbs are created by the scattering of light from the spongy matrix of keratin and air in the medullary layer of the barbs1,5. However, the precise physical mechanism for the production of these colours is still controversial1,3,4,6. Here we use a two-dimensional (2D) Fourier analysis of the spatial variation...

The classical transmission problem for the scattering of time-harmonic electromagnetic waves by a dielectric object is solved using layer potential techniques. The surface of the object is only assumed to be a Lipschitz manifold. Unlike in the case of smoother surfaces, neither the calculus of pseudodifferential operators nor the compactness...

This paper deals with the nonlinear approximation of band limited signals using their sampled values on an appropriate discrete set. Some spaces of sequences introduced by the author in previous work are used to measure the oscillation of such signals. These spaces can be viewed as discrete versions of Sobolev or, more generally, Beosv spaces. A di...

We prove boundedness of pseudodifferential operators with symbols satisfying the conditions j@ fi ¸ @ fl x a(x; ¸)j C fi;fl j¸j mGammajfij+jflj on homogeneous Besov-Lipschitz and Triebel-Lizorkin spaces 1 Introduction The study of pseudodifferential operators with symbols in the exotic classes S m 1;1 has received a lot of attention. These are oper...

The oscillatory behavior of functions with compactly supported Fourier transform is characterized in a quantified way using various function spaces. In particular, the results in this paper show that the oscillations of a function at large scale are comparable to the oscillations of its samples on an appropriate discrete set of points. Several open...

Layer-potential techniques are used to study a transmission problem arising in the scattering of electromagnetic waves by a penetrable object. The method proposed does not involve the use of the calculus of pseudodifferential operators and hence it can be applied in domains with very little regularity. The solutions are represented as a combination...

We use the method of layer potentials to study a transmission problem for the Helmholtz equation in Lipschitz domains. Following the approach for the case of smooth interface, we propose as solution an ansatz of combinations of single and double layer potentials. In the case of Lipschitz interfaces, however, the lack of smoothness introduced some a...

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Calderón-Zygmund operators are generalizations of the singular integral operators introduced by Calderón and Zygmund in the fifties [CZ]. These singular integrals are principal value convolutions of the form Tf(x) = líme®0 ò|x-y|>e K(x-y) f(y) dy = p.v.K * f(x), where f belongs to some class of test functions.

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the H\"ormander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic exp...

A systematic treatment of multilinear Calderr on-Zygmund operators is presented. The theory developed includes strong type and endpoint weak type esti-mates, interpolation, the multilinear T1 theorem, and a variety of results regarding multilinear multiplier operators.

We present some sampling theorems for functions with Fourier transform supported in a B-transversal set. These results generalize the classical Shannon sampling theorem. As applications, we obtain the φ-transform identity and a simple orthonormal wavelet system, called the Shannon wavelets. We obtain these results in each of the following settings:...