Sup-norm Estimates Research Articles

Solving the Kerzman’s problem on the sup-norm estimate for \overline{∂} on product domains

In this paper, the author solves the long term open problem of Kerzman on sup-norm estimate for Cauchy-Riemann equation on polydisc in n n -dimensional complex space. The problem has been open since 1971. He also extends and solves the problem on product domains Ω n \Omega ^n , where Ω \Omega is any bounded domain in C \mathbb {C} with C 1 , α C^{1,\alpha } boundary for some α > 0 \alpha >0 .

Large Time Decay Estimates of the Solution to the Cauchy Problem of Doubly Degenerate Parabolic Equations with Damping

In this paper we study the large time behaviour of the solution and compactification of support to the Cauchy problem for doubly degenerate parabolic equations with strong gradient damping. Under the suitable assumptions on the structure of the equation and data of the problem we establish new sharp bound of solutions for a large time. Moreover, when the support of initial datum is compact we prove that the support of the solution contains in~the~ball with radius which is independent in~time variable. In the critical case of the behaviour of the damping term the support of the solution depends on time variable logarithmically for a sufficiently large time. The main tool of the proof is based on nontrivial use of cylindrical Gagliardo--Nirenberg type embeddings and recursive inequalities. The sup-norm estimates of the solution is carried out by modified version of the classical method of De-Giorgi--Ladyzhenskaya--Uraltseva--DiBenedetto. The approach of the paper is flexible enough and can be used when studying the Cauchy--Dirichlet or Cauchy--Neumann problems in domains with non compact boundaries.

Caustics of Weakly Lagrangian Distributions

We study semiclassical sequences of distributions \(u_h\) associated with a Lagrangian submanifold of phase space \(\mathcal {L}\subset T^*X\). If \(u_h\) is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on \(\mathcal {L},\) then the asymptotics of \(u_h\) are well understood by work of Arnol’d, provided \(\mathcal {L}\) projects to X with a stable simple Lagrangian singularity. We establish sup-norm estimates on \(u_h\) under much more general hypotheses on the rate at which it is concentrating on \(\mathcal {L}\) (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.

Convolution powers of complex functions on $\mathbb Z^d$

The study of convolution powers of a finitely supported probability distribution \phi on the d -dimensional square lattice is central to random walk theory. For instance, the n th convolution power \phi^{(n)} is the distribution of the n th step of the associated random walk and is described by the classical local limit theorem. Following previous work of P. Diaconis and the authors, we explore the more general setting in which \phi takes on complex values. This problem, originally motivated by the problem of Erastus L. De Forest in data smoothing, has found applications to the theory of stability of numerical difference schemes in partial differential equations. For a complex valued function \phi on \mathbb{Z}^d , we ask and address four basic and fundamental questions about the convolution powers \phi^{(n)} which concern sup-norm estimates, generalized local limit theorems, pointwise estimates, and stability. This work extends one-dimensional results of I.J. Schoenberg, T.N.E. Greville, P. Diaconis and the second author and, in the context of stability theory, results by V. Thomée and M.V. Fedoryuk.

Dirac spectral flow on contact three manifolds I: Eigensection estimates and spectral asymmetry

Let Y be a compact, oriented 3-manifold with a contact form a and a metric ds. Suppose that F → Y is a principal bundle with structure group U(2) = SU(2) ×{±1} S such that F/S is the principal SO(3) bundle of orthonormal frames for TY . A unitary connection A0 on the Hermitian line bundle F ×det U(2) C determines a self-adjoint Dirac operator D0 on the C-bundle F ×U(2) C. The contact form a can be used to perturb the connection A0 by A0− ira. This associates a one parameter family of Dirac operators Dr for r ≥ 0. When r >> 1, we establish a sharp supnorm estimate on the eigensections of Dr with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from D0 to Dr. In particular, it implies that the subleading order term of the spectral flow is strictly smaller than O(r 3 2 ). We also relate the η-invariant of Dr to certain spectral asymmetry function involving only the small eigenvalues of Dr.

Improved critical eigenfunction estimates on manifolds of nonpositive curvature

We prove new improved endpoint, $L^{p_c}$, $p_c=\tfrac{2(n+1)}{n-1}$, estimates (the kink point) for eigenfunctions on manifolds of nonpositive curvature. We do this by using energy and dispersive estimates for the wave equation as well as new improved $L^p$, $2<p< p_c$, bounds of Blair and the author \cite{BSTop}, \cite{BSK15} and the classical improved sup-norm estimates of Berard~\cite{Berard}. Our proof uses Bourgain's \cite{BKak} proof of weak-type estimates for the Stein-Tomas Fourier restriction theorem \cite{Tomas}--\cite{Tomas2} as a template to be able to obtain improved weak-type $L^{p_c}$ estimates under this geometric assumption. We can then use these estimates and the (local) improved Lorentz space estimates of Bak and Seeger~\cite{BakSeeg} (valid for all manifolds) to obtain our improved estimates for the critical space under the assumption of nonpositive sectional curvatures.

Focal points and sup-norms of eigenfunctions

If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order o(\lambda) saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point x_0 \in M for the geodesic flow at which the associated Perron–Frobenius operator U_{x_0}\colon L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant L^2 function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.

The Global Nonlinear Stability of Minkowski Space for Self-gravitating Massive Fields

The Hyperboloidal Foliation Method (introduced by the authors in 2014) is extended here and applied to the Einstein equations of general relativity. Specifically, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields, while existing methods only apply to massless scalar fields. First of all, by analyzing the structure of the Einstein equations in wave coordinates, we exhibit a nonlinear wave-Klein–Gordon model defined on a curved background, which is the focus of the present paper. For this model, we prove here the existence of global-in-time solutions to the Cauchy problem, when the initial data have sufficiently small Sobolev norms. A major difficulty comes from the fact that the class of conformal Killing fields of Minkowski space is significantly reduced in the presence of a massive scalar field, since the scaling vector field is not conformal Killing for the Klein–Gordon operator. Our method relies on the foliation (of the interior of the light cone) of Minkowski spacetime by hyperboloidal hypersurfaces and uses Lorentz-invariant energy norms. We introduce a frame of vector fields adapted to the hyperboloidal foliation and we establish several key properties: Sobolev and Hardy-type inequalities on hyperboloids, as well as sup-norm estimates, which correspond to the sharp time decay for the wave and the Klein–Gordon equations. These estimates allow us to control interaction terms associated with the curved geometry and the massive field by distinguishing between two levels of regularity and energy growth and by a successive use of our key estimates in order to close a bootstrap argument.

An asymptotic supnorm estimate for solutions of 1-D systems of convection–diffusion equations

We show how to use the Lp–Lq approach to obtain fundamental estimates for the spatial supnorm values of solutions u(x,t)=(u1(x,t),…,um(x,t)) to general systems of convection–diffusion equations of the form∂ui∂t+∂∂xfi(x,t,u1,…,um)+∂∂xgi(t,ui)=μi(t)∂2ui∂x2,1≤i≤m, with initial data u(⋅,0)∈Lp0(R)∩L∞(R) for some 1≤p0<∞, where μi(t)>0, given arbitrary f=(f1,…,fm), g=(g1,…,gm) such that |f(x,t,u)|≤B(t)|u| for all x∈R, t≥0, u∈Rm, where B∈C0([0,∞[).

General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

We derive general bounds for the large time size of supnorm values ∥u(·,t)∥L∞(ℝ) of solutions to one-dimensional advection-diffusion equations ut+(b(x,t)u)x=uxx,x∈ℝ,t&gt;0 with initial data u(·,0)∈Lp0(ℝ)∩L∞(ℝ) for some 1≤p0&lt;∞ and arbitrary bounded advection speeds b(x,t), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.

On some energy inequalities and supnorm estimates for advection–diffusion equations in [formula omitted

We show how some suitable standard energy inequalities can be used for a simple short derivation of the fundamental supnorm estimate ‖u(⋅,t)‖L∞(Rn)≤K(n,p)‖u(⋅,0)‖Lp(Rn)t−n2p∀t>0 for solutions of advection–diffusion equations of the form ut+divf(u)=Δu (and some generalizations) in the whole space Rn, where n≥1 is arbitrary. Our approach can be used for other parabolic equations as well, including degenerate cases like porous medium type equations and p-Laplacian evolution equations. Some open problems and related results of interest are also given.

New estimates for the minimal L2 solution of ∂¯$\bar{\partial }$ and applications to geometric function theory in weighted Bergman spaces

Abstract. We introduce new twisting techniques to establish new estimates for the minimal L2 solution of ∂ ¯ $\bar{\partial }$ on weighted Bergman spaces in the unit ball. The estimate is used to obtain off-diagonal decay bounds for the associated Bergman kernel, and also improve sup-norm estimates obtained some time ago by Berndtsson.

On Supnorm Estimates for $\bar{\partial}$ on Infinite Type Convex Domains in ℂ2

In this paper, we study the \(\bar{\partial}\) equation on some convex domains of infinite type in ℂ2. In detail, we prove that supnorm estimates hold for infinite exponential type domains, provided the exponent is less than 1.

Sup-norm estimates for Bergman-projections on regulated domains

We give sufficient and necessary conditions for the boundedness of generalized Bergman projections on the space $L^\infty_v(\Omega)$. The conditions depend on the geometry of the simply connected domain $\Omega \subset \mathsf C$.

MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION

The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region. Finally, a few examples of application are given.

The explicit solutions of $$C_{p,q}^{k + \alpha } $$ -form for the $$\bar \partial $$ -equations on pseudoconvex open sets

In this paper, we study the integral solution operators for the\(\bar \partial \)-equations on pseudoconvex domains. As a generalization of [1] for the\(\bar \partial \)-equations on pseudoconvex domains with boundary of classC∞, we obtain the explicit integral operator solutions of\(C_{p,q}^{k + \alpha } \)-form for the\(\bar \partial \)-equations on pseudoconvex open sets with boundary ofC k (k≥0) and the sup-norm estimates of which solutions have similar as that [1] in form.

The sharpness of a pointwise error bound in connection with linear finite elements

Given a variational formulation of the two-dimensional Poisson problem, the present paper studies the error in approximating the exact solution via the finite element method baised upon linear elements. Using K-functional techniques, a direct sup-norm estimate in terms of a modulus of continuity of the exact solution follows directly from a well-known Jackson-type result. Because of the singularity of the Green's function associated the error bound contains a typical logarithm factor. Indeed, R. Haverkamp (1984) hats shown that this log-factor is necessary. His proof employs properties of the discrete Green's function for a related five point difference discretization. We adopt this method to establish the pointwise sharpness of the intermediate error estimate on a dense set in connection with general Lipschitz classes. The basic tool additionally used is a quantitative extension of the uniform boundedness principle. Mathematics Subject Classifications 1991: 41A25, 65N15, 65N30

Sup-norm estimates in Glimm's method

We give a proof that Total Variation { u 0(·)} ⪡ 1 can be replaced by Sup{ u 0(·) ⪡ 1 in Glimm's method whenever a coordinate system of Riemann invariants is present. The argument is somewhat simpler but in the same spirit as that given by Glimm in his celebrated paper of 1965.

Exact regularity of the Bergman and Szeg� projections on domains with partially transverse symmetries

IfD is a smooth bounded pseudoconvex domain in Cn that has symmetries transverse on the complement of a compact subset of the boundary consisting of points of finite type, then the Bergman projection forD maps the Sobolev spaceWr(D) continuously into itself and the Szego projection maps the Sobolev spaceWsur(bD) continuously into itself. IfD has symmetries, coming from a group of rotations, that are transverse on the complement of aB-regular subset of the boundary, then the Bergman projection, the Szego projection, and the\(\bar \partial \)-Neumann operator on (0, 1)-forms all exactly preserve differentiability measured in Sobolev norms. The results hold, in particular, for all smooth bounded strictly complete pseudoconvex Hartogs domains in C2, as well as for Sibony's counterexample domain that fails to have sup-norm estimates for solutions of the\(\bar \partial \)-equation.

Finitely generated ideals in $A(\omega )$

The Gleason problem is solved on real analytic pseudoconvex domains in C 2 . In this case the weakly pseudoconvex points can be a two-dimensional subset of the boundary. To reduce the Gleason problem to a ∂ ¯ question it is shown that the set of Kohn-Nirenberg points is at most one-dimensional. In fact, except for a one-dimensional subset, the weakly pseudoconvex boundary points are R-points as studied by Range and therefore allow local sup-norm estimates for ∂ ¯.