# Browsing by Subject "OPERATORS"

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Now showing items 1-12 of 12
• (2019)
Let mu be Radon measures on R, with mu nonatomic and nu doubling, and write mu = mu(a) + mu(s) for the Lebesgue decomposition of mu relative to nu. For an interval I subset of R, define alpha(mu,nu) (I) := W-1 (mu(I), nu(I)), the Wasserstein distance of normalised blow-ups of mu and nu restricted to I. Let S nu be the square function S-nu(2) (mu) = Sigma alpha(2)(mu,nu)(I) chi(1), where D is the family of dyadic intervals of side-length at most 1. I prove that S-nu(mu) is finite mu(a) almost everywhere and infinite mu(s) almost everywhere. I also prove a version of the result for a nondyadic variant of the square function S-nu(mu). The results answer the simplest "n = d = 1" case of a problem of J. Azzam, G. David and T. Toro.
• (2019)
We represent a general bilinear Calderon-Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so-called sparse T1 theorem for bilinear singular integrals.
• (2019)
We study singular integral operators induced by 3-dimensional Calderon-Zygmund kernels in the Heisenberg group. We show that if such an operator is L (2) bounded on vertical planes, with uniform constants, then it is also L-2 bounded on all intrinsic graphs of compactly supported C-1,C-alpha functions over vertical planes. In particular, the result applies to the operator R, induced by the kernel K(z) = del(H )parallel to z parallel to(-2), z is an element of H \ {0}, the horizontal gradient of the fundamental solution of the sub-Laplacian. The L-2 boundedness of R, is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of non-removable sets with positive and locally finite 3-dimensional measure. (C) 2019 Elsevier Inc. All rights reserved.
• (2018)
We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are: Passive observations in spacetime: consider observations in an open set . The light observation set corresponding to a point source at is the intersection of V and the light-cone emanating from the point q. Let be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points determine uniquely the conformal type of W. Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from and return back to mu.
• (2018)
We consider inverse problems in space-time (M, g), a 4-dimensional Lorentzian manifold. For semilinear wave equations square(g)u + H (x, u) = f, where square(g) denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map , L : f -> u broken vertical bar v, where V is a neighborhood of a time-like geodesic mu, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set, where waves can propagate from mu and return back. Moreover, on a given space-time (M, g), the source-to-solution map determines some coefficients of the Taylor expansion of H in u.
• (2020)
We prove L-p bounds for the extensions of standard multilinear Calderon-Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space-in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative L-p spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.
• (2018)
In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev-Poincar, inequality. We show that this method gives a Sobolev-Poincar, inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev-Poincar, inequality depends on the domain. The Sobolev-Poincar, inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy-Littlewood maximal operator.
• (2018)
We determine the solid hull and solid core of weighted Banach spaces H-upsilon(infinity) of analytic functions functions f such that upsilon vertical bar f vertical bar is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights upsilon. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if H-upsilon(infinity) is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in H-upsilon(infinity). As a consequence H-upsilon(infinity) does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.
• (2018)
Let v(r) = exp( a/(1 r)(b)) with a > 0 and 0 <b
• (2018)
Given a continuous, radial, rapidly decreasing weight v on the complex plane, we study the solid hull of its associated weighted space H-v(infinity)(C) of all the entire functions f such that v vertical bar f vertical bar is bounded. The solid hull is found for a large class of weights satisfying the condition ( B) of Lusky. Precise formulations are obtained for weights of the form v( r) = exp(-ar(p)), a > 0, p > 0. Applications to spaces of multipliers are included.
• (2018)
We introduce a fundamentally new method for the design of metamaterial arrays. These behave superdimensionally, exhibiting a higher local density of resonant frequencies, giant focusing of rays, and stronger concentration of waves than expected from the physical dimension. This sub-Riemannian optics allows planar designs to function effectively as 3- or higher-dimensional media, and bulk material as dimension 4 or higher. Valid for any waves modeled by the Helmholtz equation, including scalar optics and acoustics, and with properties derived from the behavior of waves in sub-Riemannian geometry, these arrays can be assembled from nonresonant metamaterial cells and are potentially broadband. Possible applications include antenna design and energy harvesting.
• (2020)
We study the weighted light ray transform L of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze L as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function f from its the weighted light ray transform Lf by a suitable filtered back-projection.