Endpoint Regularity Research Articles

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Global endpoint regularity estimates for the fractional Dirichlet problem

Global endpoint regularity estimates for the fractional Dirichlet problem

Endpoint regularity of discrete multilinear maximal and minimal operators

We introduce two discrete multilinear maximal and minimal operators associated to the function [Formula: see text], which cover the classical discrete multilinear maximal and minimal operators and their fractional variants. Under a more restrictive condition on [Formula: see text], we establish some new variation boundedness and continuity of the above operators, which is new and interesting even in the linear case. It should be pointed out that the main results we obtain not only extend and improve some known ones in the maximal setting, but also supplement some new results in the minimal setting.

Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations

It has recently been proposed by the author of the present work that the periodic NS equations (PNS) with high energy assumption can breakdown in finite time but with sufficient low energy scaling the equations may not exhibit finite time blowup.This article gives a general model using specific periodic special functions, that is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the PNS equations at the centers of cells of the 3-Torus. Satisfying a divergence free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which blowup in finite time for PNS can occur starting with the first derivative and higher with respect to time. P. Isett (2016) has shown that the conservation of energy fails for the 3D incompressible Euler flows with Hlder regularity below 1/3.(Onsager’s second conjecture) The endpoint regularity in Onsager’s conjecture is addressed, and it is found that conservation of energy occurs when the Hlder regularity is exactly 1/3. The endpoint regularity problem has important connections with turbulence theory. Finally very recent developed new governing equations of fluid mechanics are proposed to have no finite time singularities.

On smoothing estimates in modulation spaces and the nonlinear Schrödinger equation with slowly decaying initial data

We show new local Lp-smoothing estimates for the Schrödinger equation in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and Lp-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in Lp(Rd) for d≥1 and p>2, which was previously known for d≥2. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schrödinger equation on the line. Lastly, we show ℓ2-decoupling inequalities for variable-coefficient versions of elliptic and non-elliptic Schrödinger phase functions.

Endpoint regularity for 2d Mumford-Shah minimizers: On a theorem of Andersson and Mikayelyan

We give an alternative proof of the regularity, up to the loose end, of minimizers, resp. critical points of the Mumford-Shah functional when they are sufficiently close to the cracktip, resp. they consist of a single arc terminating at an interior point.

Endpoint Regularity of the Discrete Multisublinear Fractional Maximal Operators

We study the endpoint regularity of the one-dimensional discrete multisublinear fractional maximal operators, both in the centered and uncentered versions. Some new variation inequalities will be proved for the above operators acting on the vector-valued function $$\mathbf {f}=(f_1, \ldots ,f_m)$$ with each $$f_j$$ belonging to $$\mathrm{BV}({\mathbb {Z}})$$ or $$\ell ^1({\mathbb {Z}})$$ , where $$\mathrm{BV}({\mathbb {Z}})$$ denotes the set of functions of bounded variation defined on $${\mathbb {Z}}$$ . In addition, it was also shown that the above operators are bounded and continuous from $$\ell ^1({\mathbb {Z}})\times \cdots \times \ell ^1({\mathbb {Z}})$$ to $$\mathrm{BV}({\mathbb {Z}})$$ . The above results represent significant and natural extensions of what was known previously.

The end-point regularity criterion for the Navier–Stokes equations in terms of [formula omitted

We show the following extension of the recent optimal regularity results by Kukavica & Ziane, Cao and Zhang: If u is a solution of the Navier–Stokes equations in the whole space R3 and ∂3u∈Lp(0,T;Lq(R3)), T>0, where 2∕p+3∕q=2 and q∈(3∕2,3], then u is regular on (0,T].

Weak differentiability for fractional maximal functions of general Lp functions on domains

Let Ω⊂Rn be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to Ω maps Lp(Ω)→W1,p(Ω) for all p>1, when the smoothness index α≥1. In particular, the results are valid in the range p∈(1,n/(n−1)] that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.

Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrodinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in \begin{document}$ H^s({\mathbb{R}}) $\end{document} , \begin{document}$ s>\frac12 $\end{document} , without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the \begin{document}$ H^{s-1}({\mathbb{R}}) $\end{document} -norm.

A note on the endpoint regularity of the discrete maximal operator

A note on the endpoint regularity of the discrete maximal operator

Endpoint regularity criterion for weak solutions of the 3D incompressible liquid crystals system

In this paper, we consider the endpoint case regularity for the 3D liquid crystals system. We prove that if , then weak solution (v,d) is smooth, and our main observation is that the condition is not necessary in this situation. The proof is based on the blow‐up analysis and backward uniqueness for the parabolic operator developed by Escauriaza‐Seregin‐S̆verák.

Endpoint regularity of discrete multisublinear fractional maximal operators associated with ell^{1}-balls

In this paper we investigate the endpoint regularity of the discrete m-sublinear fractional maximal operator associated with ell^{1}-balls, both in the centered and uncentered versions. We show that these operators map ell^{1}(mathbb{Z}^{d})timescdotstimes ell^{1}(mathbb{Z}^{d}) into operatorname{BV}(mathbb{Z}^{d}) boundedly and continuously. Here operatorname{BV}(mathbb{Z}^{d}) represents the set of functions of bounded variation defined on mathbb{Z}^{d}.

Endpoint Regularity of Multisublinear Fractional Maximal Functions

AbstractIn this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

Some remarks on the $L^p$ regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition

In this note we prove an end-point regularity result on the L^p integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations. These results are analogous to corresponding results for divergence form elliptic equations in [3, 15].

A NOTE ON THE ENDPOINT REGULARITY OF THE HARDY–LITTLEWOOD MAXIMAL FUNCTIONS

In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$. Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.

Sets of Salem type and sharpness of the 𝐿²-Fourier restriction theorem

We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the L 2 L^2 -Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M s p , q and Wiener amalgam spaces W s p , q . We show that the periodic Brownian motion belongs locally in time to M s p , q ( T ) and W s p , q ( T ) for ( s − 1 ) q < − 1 , and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T , we show that ( M s p , q ( T ) , μ ) and ( W s p , q ( T ) , μ ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space b ˆ p , ∞ s ( T ) . Specifically, we prove that the Brownian motion belongs to b ˆ p , ∞ s ( T ) for ( s − 1 ) p = − 1 , and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces B p , q s , and indicate the endpoint large deviation estimates.

Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity H -¾ , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [- T, T ] and a locally Lipschitz continuous map taking the initial data in H -¾ to a distributional solution u ∈ C 0 ([- T, T ]; H -¾ ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.