Browsing by Author "Spector, Daniel"

Now showing 1 - 18 of 18

A Decomposition by Non-Negative Functions in the Sobolev Space W-k,W-1
(2020-01-01)
Ponce, Augusto C.
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
We show how a strong capacitary inequality can be used to give a decomposition of any function in the Sobolev space W-k,W-1 (R-d) as the difference of two non-negative functions in the same space with control of their norms.
A noninequality for the fractional gradient
(2019-01-01)
Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper we give a streamlined proof of an inequality recently obtained by the author: For every alpha is an element of (0, 1) there exists a constant C = C(alpha, d) > 0 such that parallel to u parallel to(Ld/(d-alpha), 1(Rd)) <= C parallel to D(alpha)u parallel to(L1(Rd; Rd)) for all u is an element of L-q(R-d) for some 1 <= q < d/(1 - alpha) such that D(alpha)u := VI(1-alpha)u is an element of L-1(R-d; R-d). We also give a counterexample which shows that in contrast to the case alpha = 1, the fractional gradient does not admit an L-1 trace inequality, i.e. parallel to D(alpha)u parallel to(L1(Rd; Rd)) cannot control the integral of u with respect to the Hausdorff content H-co(d-alpha). The main substance of this counter-example is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space L-1(H-co(d-beta)), beta is an element of [1, d). It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to beta is an element of (0, 1).
A note on estimates for elliptic systems with L-1 data
(2019-11-01)
Raita, Bogdan
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper, we give necessary and sufficient conditions on the compatibility of a kth-order homogeneous linear elliptic differential operator A and differential constraint C for solutions to Au = f subject to Cf = 0 in R-n to satisfy the estimates parallel to D(k-j)u parallel to(Ln/n-j(Rn)) <= c parallel to f parallel to(L1(Rn)) for j is an element of {1, ..., min{k, n-1}} and parallel to D(k-n)u parallel to(L infinity(Rn)) <= c parallel to f parallel to(L1(Rn)) when k >= n. (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
A note on the fractional perimeter and interpolation
(2017-09-01)
Ponce, Augusto C.
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces W-alpha,W-1 of order 0 < alpha < 1. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
A remark on an integral characterization of the dual of BV
(2018-01-15)
Fusco, Nicola
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper, we show how under the continuum hypothesis one can obtain an integral representation for elements of the topological dual of the space of functions of bounded variation in terms of Lebesgue and Kolmogorov-Burkill integrals. (C) 2017 Elsevier Inc. All rights reserved.
An L-1-type estimate for Riesz potentials
(2017-01-01)
Schikorra, Armin
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Spector, Daniel
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Van Schaftingen, Jean
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應用數學系
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Department of Applied Mathematics
In this paper we establish new L-1-type estimates for the classical Riesz potentials of order alpha is an element of (0, N): parallel to I alpha u parallel to N-L/(N-alpha)(R-N) <= C parallel to Ru parallel to (L1(RN; RN)). This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space H-1 (R-N) and provides a new family of L-1-Sobolev inequalities for the Riesz fractional gradient.
An optimal Sobolev embedding for L-1
(2020-08-15)
Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant C = C(alpha, d) > 0 such that parallel to L alpha F parallel to(Ld/(d-alpha),1(Rd;Rd)) <= C parallel to F parallel to(L1(Rd;Rd)) for all fields F is an element of L-1 (R-d;R-d) such that curl F = 0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime p = 1 of the well-established results for p > 1. (C) 2020 The Author. Published by Elsevier Inc.
Analysis and Application of a Nonlocal Hessian
(2015-01-01)
Lellmann, Jan
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Papafitsoros, Konstantinos
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Schoenlieb, Carola
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Spector, Daniel
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應用數學系
In this work we introduce a formulation for a nonlocal Hessian that combines the ideas of higher-order and nonlocal regularization for image restoration, extending the idea of nonlocal gradients to higher-order derivatives. By intelligently choosing the weights, the model allows us to improve on the current state of the art higher-order method, total generalized variation, with respect to overall quality and preservation of jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights it can be shown to converge to classical second-order regularizers, and in fact it allows a novel characterization of higher-order Sobolev and BV spaces.
L On formulae decoupling the total variation of BV functions
(2017-05-01)
Ponce, Augusto C.
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function u is an element of SBV converge strictly to the singular portion of D-u. (C) 2016 Elsevier Ltd. All rights reserved.
L-p theory for fractional gradient PDE with VMO coefficients
(2015-01-01)
Schikorra, Armin
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Shieh, Tien-Tsan
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Spector, Daniel
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交大名義發表
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National Chiao Tung University
In this paper, we prove L-p estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are VMO. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.
L-p-Taylor approximations characterize the Sobolev space W-1,W-p
(2015-04-01)
Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this note, we introduce a variant of Calderon and Zygmund\'s notion of L-p-differentiability - an L-p-Taylor approximation. Our first result is that functions in the Sobolev space W-1,W-p(RN) possess a first-order L-p-Taylor approximation. This is in analogy with Calderon and Zygmund\'s result concerning the L-p-differentiability of Sobolev functions. In fact, the main result we announce here is that the first-order L-p-Taylor approximation characterizes the Sobolev space W-1,W-p(RN), and therefore implies L-p-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderon-Zygmund classes with others due to Bourgain, Brezis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising; however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space. (C) 2015 Published by Elsevier Masson SAS on behalf of Academie des sciences.
New directions in harmonic analysis on L-1
(2020-03-01)
Spector, Daniel
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應用數學系
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Department of Applied Mathematics
The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there are still things we do not understand about the action of integral operators on functions. This is no more apparent than in the L-1 setting, where only recently have optimal inequalities been obtained on the Lebesgue and Lorentz scale for scalar functions, while the full resolution of similar estimates for vector-valued functions is incomplete. The purpose of this paper is to discuss how some often overlooked estimates for the classical Poisson equation give an entry into these questions, to present the state of the art of what is known, and to survey some open problems on the frontier of research in the area. (C) 2019 Elsevier Ltd. All rights reserved.
On a generalization of L-p-differentiability
(2016-06)
Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper we connect Calderon and Zygmund\'s notion of L-p-differentiability (Calderon and Zygmund, Proc Natl Acad Sci USA 46: 1385-1389, 1960) with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu (Optimal Control and Partial Differential Equations, pp. 439-455, 2001). We showhowthe results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev space W-1,W-1, and complete the proof of a characterization of the Sobolev spaces recently claimed in (Leoni and Spector, J Funct Anal 261: 2926-2958, 2011; Leoni and Spector, J Funct Anal 266: 1106-1114, 2014).
On the regularity of solutions to Poisson's equation
(2015-09-01)
Garg, Rahul
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson\'s equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In particular, our results show that there is no qualitative difference in the regularity of these solutions in the plane and in higher dimensions. (C) 2015 Published by Elsevier Masson SAS on behalf of Academie des sciences.
On the Role of Riesz Potentials in Poisson\'s Equation and Sobolev Embeddings
(2015)
Garg, Rahul
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper, we show how standard techniques can be used to obtain new "almost"-Lipschitz estimates for the classical Riesz potentials acting on L-P-spaces in the supercritical exponent. Whereas similar results are known to hold for Riesz potentials acting on L-P(Omega) for Omega subset of R-N a bounded domain (and also on Sobolev, Sobolev-Orlicz functions), our results concern the mapping properties of the Riesz potentials on all of R-N. Additionally, we introduce and prove sharp estimates on the modulus of continuity for a family of Riesz-type potentials. In particular, through a new representation via these Riesz-type potentials, we establish analagous results for the logarithmic potential. As applications of these continuity estimates, we deduce new regularity estimates for distributional solutions to Poisson\'s equation, as well as an alternative proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
On the Role of Riesz Potentials in Poisson\'s Equation and Sobolev Embeddings
(2015-01-01)
Garg, Rahul
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper, we show how standard techniques can be used to obtain new "almost"-Lipschitz estimates for the classical Riesz potentials acting on L-P-spaces in the supercritical exponent. Whereas similar results are known to hold for Riesz potentials acting on L-P(Omega) for Omega subset of R-N a bounded domain (and also on Sobolev, Sobolev-Orlicz functions), our results concern the mapping properties of the Riesz potentials on all of R-N. Additionally, we introduce and prove sharp estimates on the modulus of continuity for a family of Riesz-type potentials. In particular, through a new representation via these Riesz-type potentials, we establish analagous results for the logarithmic potential. As applications of these continuity estimates, we deduce new regularity estimates for distributional solutions to Poisson\'s equation, as well as an alternative proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma
(2019-01-01)
Spector, Daniel
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Van Schaftingen, Jean
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應用數學系
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Department of Applied Mathematics
We prove a family of Sobolev inequalities of the form parallel to u parallel to(n/Ln-1, 1(Rn, V)) <= C parallel to A(D)u parallel to(L1(Rn,E)) where A(D) : C-c(infinity)(R-n ,V) -> C-c(infinity)(R-n , E) is a vector first-order homogeneous linear differential operator with constant coefficients, a is a vector field on R-n and L-n/n-1,L- 1 (R-n) is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo-Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis-Whitney inequality and Gagliardo's lemma.
SOME REMARKS ON BOUNDARY OPERATORS OF BESSEL EXTENSIONS
(2018-06-01)
Goodman, Jesse
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Spector, Daniel
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應用數學系
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Department of Applied Mathematics
In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is Delta(x)u(x,y) +(1-2s)(y) (partial derivative u)(partial derivative y)(x,y) + (2)(partial derivative y) (partial derivative 2u)(x,y) = 0 for x is an element of R-d,y > 0, u(x,0) = f(x) for x is an element of R-d. In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases s = k is an element of N.