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Journal articles on the topic 'Logarithmic Sobolev spaces'

Author: Grafiati

Published: 14 March 2023

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1

Ghobber, Saifallah, and Hatem Mejjaoli. "Logarithm Sobolev and Shannon’s Inequalities Associated with the Deformed Fourier Transform and Applications." Symmetry 14, no. 7 (June 24, 2022): 1311. http://dx.doi.org/10.3390/sym14071311.

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By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash’s inequality, Carlson’s inequality and Sobolev’s embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the L∞-norm. Finally, we obtain a logarithmic Sobolev inequality in Lp-spaces, from which we derive an Lp-Heisenberg-type uncertainty inequality and an Lp-Nash-type inequality for the Dunkl transform.

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2

Marton, Katalin. "Logarithmic Sobolev inequalities in discrete product spaces." Combinatorics, Probability and Computing 28, no. 06 (June 13, 2019): 919–35. http://dx.doi.org/10.1017/s0963548319000099.

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AbstractThe aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*) $$D(p||q){\rm{\le}}C \cdot \sum\limits_{i = 1}^n {{\rm{\mathbb{E}}}_p D(p_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,...,{\rm{ }}Y_n )||q_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,{\rm{ }}...,{\rm{ }}Y_n )),} $$ where pi(· |y1, …, yi−1, yi+1, …, yn) and qi(· |x1, …, xi−1, xi+1, …, xn) denote the local specifications for p resp. q, that is, the conditional distributions of the ith coordinate, given the other coordinates. The constant C depends on (the local specifications of) q.The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy.In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.

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3

Chaabane, Slim, and Imed Feki. "Optimal logarithmic estimates in Hardy–Sobolev spaces." Comptes Rendus Mathematique 347, no. 17-18 (September 2009): 1001–6. http://dx.doi.org/10.1016/j.crma.2009.07.018.

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4

Edmunds, D. E., and H. Triebel. "Logarithmic Sobolev Spaces and Their Applications to Spectral Theory." Proceedings of the London Mathematical Society s3-71, no. 2 (September 1995): 333–71. http://dx.doi.org/10.1112/plms/s3-71.2.333.

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5

Hsu, Elton P. "Logarithmic Sobolev Inequalities on Path Spaces Over Riemannian Manifolds." Communications in Mathematical Physics 189, no. 1 (October 1, 1997): 9–16. http://dx.doi.org/10.1007/s002200050188.

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6

Gressman, Philip T. "Fractional Poincaré and logarithmic Sobolev inequalities for measure spaces." Journal of Functional Analysis 265, no. 6 (September 2013): 867–89. http://dx.doi.org/10.1016/j.jfa.2013.05.036.

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7

Ibrahim, H. "A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class." Journal of Function Spaces and Applications 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/148706.

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In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brézis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

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8

Ehler, Martin, Manuel Gräf, and Chris J. Oates. "Optimal Monte Carlo integration on closed manifolds." Statistics and Computing 29, no. 6 (October 30, 2019): 1203–14. http://dx.doi.org/10.1007/s11222-019-09894-w.

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Abstract The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $$n^{-1/2}$$n-1/2. However, the re-weighting of random points, as exemplified in the Bayesian Monte Carlo method, can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor. We also provide numerical experiments matching the theoretical results for some Sobolev spaces on the sphere $${\mathbb {S}}^2$$S2 and on the Grassmannian manifold $${\mathcal {G}}_{2,4}$$G2,4. Our theoretical findings also cover function spaces on more general sets such as the unit ball, the cube, and the simplex.

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9

Aouaoui, Sami, and Rahma Jlel. "A new Singular Trudinger–Moser Type Inequality with Logarithmic Weights and Applications." Advanced Nonlinear Studies 20, no. 1 (February 1, 2020): 113–39. http://dx.doi.org/10.1515/ans-2019-2068.

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AbstractIn this paper, we establish a new singular Trudinger–Moser type inequality for radial Sobolev spaces with logarithmic weights. The existence of nontrivial solutions is proved for an elliptic equation defined in {\mathbb{R}^{n}}, relying on variational methods and involving a nonlinearity with doubly exponential growth at infinity.

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10

Machihara, Shuji, Tohru Ozawa, and Hidemitsu Wadade. "Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces." Journal of Inequalities and Applications 2013, no. 1 (2013): 381. http://dx.doi.org/10.1186/1029-242x-2013-381.

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11

Li, Pengtao, Qixiang Yang, and Yueping Zhu. "Wavelets, Sobolev Multipliers, and Application to Schrödinger Type Operators with Nonsmooth Potentials." Abstract and Applied Analysis 2013 (2013): 1–22. http://dx.doi.org/10.1155/2013/193420.

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We employ Meyer wavelets to characterize multiplier spaceXr,pt(ℝn)without using capacity. Further, we introduce logarithmic Morrey spacesMr,pt,τ(ℝn)to establish the inclusion relation between Morrey spaces and multiplier spaces. By fractal skills, we construct a counterexample to show that the scope of the indexτofMr,pt,τ(ℝn)is sharp. As an application, we consider a Schrödinger type operator with potentials inMr,pt,τ(ℝn).

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12

Aida, Shigeki. "Logarithmic Derivatives of Heat Kernels and Logarithmic Sobolev Inequalities with Unbounded Diffusion Coefficients on Loop Spaces." Journal of Functional Analysis 174, no. 2 (July 2000): 430–77. http://dx.doi.org/10.1006/jfan.2000.3592.

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13

Marton, Katalin. "An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces." Journal of Functional Analysis 264, no. 1 (January 2013): 34–61. http://dx.doi.org/10.1016/j.jfa.2012.10.001.

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14

Inahama, Yuzuru. "Logarithmic Sobolev Inequality on Free Loop Groups for Heat Kernel Measures Associated with the General Sobolev Spaces." Journal of Functional Analysis 179, no. 1 (January 2001): 170–213. http://dx.doi.org/10.1006/jfan.2000.3677.

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15

Capitaine, Mireille, Elton Hsu, and Michel Ledoux. "Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces." Electronic Communications in Probability 2 (1997): 71–81. http://dx.doi.org/10.1214/ecp.v2-986.

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16

Mañas-Mañas, Juan F., Juan J. Moreno-Balcázar, and Richard Wellman. "Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials." Mathematics 8, no. 2 (February 3, 2020): 182. http://dx.doi.org/10.3390/math8020182.

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In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential operator, we are interested in the corresponding eigenvalues, more exactly, in their asymptotic behavior. Thus, we can determine a limit value which links this asymptotic behavior and the uniform norm of the orthonormal polynomials in a logarithmic scale. This value appears in the theory of reproducing kernel Hilbert spaces. On the other hand, we tackle a more general case than the one considered in the literature previously.

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17

Mezouar, Nadia, Salah Mahmoud Boulaaras, and Ali Allahem. "Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms." Complexity 2020 (April 4, 2020): 1–25. http://dx.doi.org/10.1155/2020/7105387.

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In this paper, a nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay term and logarithmic nonlinearity in the weakly nonlinear internal feedback is considered, where the global and local existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo-Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. These results extend and improve many results in the literature.

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18

Choquet, C., and L. Pankratov. "Homogenization of a class of quasilinear elliptic equations with non-standard growth in high-contrast media." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 140, no. 3 (May 21, 2010): 495–539. http://dx.doi.org/10.1017/s0308210509000985.

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We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces Lpε (·) in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions pε converge uniformly to a limit function p0 and that p0 satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.

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19

Flemming, Katharina, and Peter Junghanns. "A fast algorithm for the numerical solution of an integral equation with logarithmic kernel." Publications de l'Institut Math?matique (Belgrade) 96, no. 110 (2014): 143–57. http://dx.doi.org/10.2298/pim1410143f.

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We describe an algorithm for the numerical solution of an integral equation of the form ? 1/? ?1?1 [(y ? x) ln |y ? x| ? h(x, y)] u(y) dy/?1?y2 = f(x), ?1 < x < 1, which is based on a collocation-quadrature method and which has the same convergence rate as this method, but only O(n log n) complexity. This integral equation turns out to be an ill-posed problem in (the best possible choice of) a pair of non-periodic Sobolev-like spaces. The present paper presents the technique, how to overcome this peculiarity in the investigation of the fast algorithm.

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20

Machihara, Shuji, Tohru Ozawa, and Hidemitsu Wadade. "Notes on the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’." Journal of Inequalities and Applications 2014, no. 1 (2014): 253. http://dx.doi.org/10.1186/1029-242x-2014-253.

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21

Deng, Yu, and Christian Zillinger. "Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics." Archive for Rational Mechanics and Analysis 242, no. 1 (July 30, 2021): 643–700. http://dx.doi.org/10.1007/s00205-021-01697-6.

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AbstractIn this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity.

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22

Feki, Imed, and Houda Nfata. "On Lp–L1 estimates of logarithmic-type in Hardy–Sobolev spaces of the disk and the annulus." Journal of Mathematical Analysis and Applications 419, no. 2 (November 2014): 1248–60. http://dx.doi.org/10.1016/j.jmaa.2014.05.042.

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23

Cheng, Xu, and Detang Zhou. "Eigenvalues of the drifted Laplacian on complete metric measure spaces." Communications in Contemporary Mathematics 19, no. 01 (November 24, 2016): 1650001. http://dx.doi.org/10.1142/s0219199716500012.

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In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

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24

Zhu, Maochun, and Lianfang Wang. "Adams’ inequality with logarithmic weights in ℝ⁴." Proceedings of the American Mathematical Society 149, no. 8 (May 18, 2021): 3463–72. http://dx.doi.org/10.1090/proc/15488.

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Trudinger-Moser inequality with logarithmic weight was first established by Calanchi and Ruf [J. Differential Equations 258 (2015), pp. 1967–1989]. The aim of this paper is to address the higher order version; more precisely, we show the following inequality sup u ∈ W 0 , r a d 2 , 2 ( B , ω ) , ‖ Δ u ‖ ω ≤ 1 ∫ B exp ⁡ ( α | u | 2 1 − β ) d x > + ∞ \begin{equation*} \sup _{u \in W_{0,rad}^{2,2}(B,\omega ),{{\left \| {\Delta u} \right \|}_\omega } \le 1} \int _B {\exp \left ( {\alpha {{\left | u \right |}^{\frac {2}{{1 - \beta }}}}} \right )} dx > + \infty \end{equation*} holds if and only if \[ α ≤ α β = 4 [ 8 π 2 ( 1 − β ) ] 1 1 − β , \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}}, \] where B B denotes the unit ball in R 4 \mathbb {R}^{4} , β ∈ ( 0 , 1 ) \beta \in \left ( {0,1} \right ) , ω ( x ) = ( log ⁡ 1 | x | ) β \omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta } or ( log ⁡ e | x | ) β {\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta } , and W 0 , r a d 2 , 2 ( B , ω ) W_{0,rad}^{2,2}(B,\omega ) is the weighted Sobolev spaces. Our proof is based on a suitable change of variable that allows us to represent the laplacian of u u in terms of the second derivatives with respect to the new variable; this method was first used by Tarsi [Potential Anal. 37 (2012), pp. 353–385].

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25

Feki, Imed, Ameni Massoudi, and Houda Nfata. "A generalization to the Hardy-Sobolev spaces $H^{k,p}$ of an $L^p$-$L^1$ logarithmic type estimate." Czechoslovak Mathematical Journal 68, no. 2 (March 28, 2018): 387–414. http://dx.doi.org/10.21136/cmj.2018.0466-16.

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26

Voitovych, Mykhailo. "Continuity of weak solutions to nonlinear fourth-order equations with strengthened ellipticity via Wolff potentials." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 33–49. http://dx.doi.org/10.37069/1683-4720-2019-33-3.

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In the present article nonlinear fourth-order equations in the divergence form with L^1-right-hand sides and the strengthened ellipticity condition on the coefficients are analyzed. Such equations, but with sufficiently regular right-hand sides, first appeared in the works of Professor I.V. Skrypnik concerning the regularity of generalized solutions for multidimensional nonlinear elliptic equations of high order. This class of equations correctly generalizes the corresponding nonlinear second-order elliptic equations with non-standard growth conditions on the coefficients, which are models for numerous physical phenomena in non-homogeneous medium. The main result of the article is a theorem on an estimation of oscillations in a ball of solutions to the given equations via the Wolff potentials of their right-hand sides. To prove this, we use the improved Kilpelainen-Maly method and pointwise potential estimates of functions related to special subclasses of Sobolev spaces, akin to the well-known De Giorgi classes. A new point is the verification that these classes contain superpositions of solutions and Moser logarithmic functions that include the Wolf potential of the right-hand side of the equation. As a corollary, a new result is obtained on the interior continuity of solutions to the equations with right-hand sides from the Kato class, which is characterized by the uniform convergence to zero of the corresponding Wolff potentials. Some important cases of fulfilling this condition are considered: the right-hand side of the equation belongs to the Morrey space with an index exceeding a certain limiting value, then the solutions are locally Holder continuous; if the right-hand side belongs to the borderline Lorentz-Zygmund classes, then the solutions are only locally continuous, but they are not Holder continuous in the domain. In the case when the summability exponents of the right-hand sides of the equations under consideration are less than the borderline values, there are examples of unbounded discontinuous solutions. These facts are exact analogues of the corresponding results in the theory of second-order elliptic equations.

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27

Ryazanov, Vladimir, and Serhii Volkov. "Caratheodory theorem about prime ends on Riemann surfaces." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 34 (April 24, 2021): 100–110. http://dx.doi.org/10.37069/1683-4720-2020-34-10.

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The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.

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28

Domínguez, Óscar, and Sergey Tikhonov. "Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations." Memoirs of the American Mathematical Society 282, no. 1393 (February 2023). http://dx.doi.org/10.1090/memo/1393.

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In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

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29

Haroske, Dorothee D. "Logarithmic Sobolev spaces on ℝn ; entropy numbers, and some applications." Forum Mathematicum 12, no. 3 (January 21, 2000). http://dx.doi.org/10.1515/form.2000.007.

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30

Bardet, Ivan, and Cambyse Rouzé. "Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates." Annales Henri Poincaré, June 14, 2022. http://dx.doi.org/10.1007/s00023-022-01196-8.

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AbstractWe generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated$$\mathbb {L}_p$$ L p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated $$\mathbb {L}_p$$ L p norms are not uniformly convex for $$1\le p \le 2$$ 1 ≤ p ≤ 2 . As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.

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31

Jolly, Michael S., Anuj Kumar, and Vincent R. Martinez. "On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces." Communications on Pure & Applied Analysis, 2021, 0. http://dx.doi.org/10.3934/cpaa.2021169.

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<p style='text-indent:20px;'>This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> is of lower singularity, i.e., <inline-formula><tex-math id="M3">\begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a logarithmic smoothing operator and <inline-formula><tex-math id="M5">\begin{document}$ \beta \in [0, 1] $\end{document}</tex-math></inline-formula>. We complete this study by considering the more singular regime <inline-formula><tex-math id="M6">\begin{document}$ \beta\in(1, 2) $\end{document}</tex-math></inline-formula>. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.</p>

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