Sharp hölder estimates for\overline \partial on ellipsoids

We solve the -equation on real and on complex ellipsoids in ^N. It is proved that the solution satisfies sharp Hölder estimates. That is, the Hölder exponent equals the reciprocal of the maximal order of contact of the boundary of the ellipsoid with complex-analytic curves.Supported by NSF grant DMS 8401273.Supported by the Netherlands' organization for the advancement of pure research ZWO.     

Sharp Lp estimates for the ∂bequation on the boundaries of real ellipsoids in Cn

In this paper we prove the best possible L^p estimates for the ?-_b -equation on the boundaries of real ellipsoids in Cⁿ, and provide examples to show why they cannot be improved.     

Sharp a priori estimates for div–curl systems in Heisenberg groups

In this paper we prove a family of inequalities for differential forms in Heisenberg groups H¹${\mathbb{H}}^1$ and H²${\mathbb{H}}^2$, that are the natural counterpart of a class of div–curl inequalities in de Rham?s complex proved by Lanzani & Stein and Bourgain & Brezis.     

Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

We study the local Szegö–Weinberger profile in a geodesic ball \(B_g(y_0,r_0)\) centered at a point \(y_0\) in a Riemannian manifold \(({\mathcal {M}},g)\) . This profile is obtained by maximizing the first nontrivial Neumann eigenvalue \(\mu _2\) of the Laplace–Beltrami Operator \(\Delta _g\) on \({\mathcal {M}}\) among subdomains of \(B_g(y_0,r_0)\) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of \({\mathcal {M}}\) at \(y_0\) . As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of \(\Delta _g\) , but additional difficulties arise due to the fact that \(\mu _2\) is degenerate in the unit ball in \(\mathbb {R}^N\) and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.     

Sharp estimates for Hübner’s upper bound function with applications

New better estimates, which are given in terms of elementary functions, for the function r → (2/π)(1 - r2)K(r)K (r) + log r appearing in Hübner's sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.     

C 1,α-regularity for electrorheological fluids in two dimensions

We prove the existence of -solutions to a system of nonlinear partial differential equations describing steady planar motions of electrorheological fluids with Dirichlet boundary conditions for inf .     

The Poincaré inequality for C 1,α-smooth vector fields

We obtain the Poincaré inequality for the equiregular Carnot-Carathéodory spaces spanned by vector fields with Hölder class derivatives.     

Sharp Estimates for Schrödinger Groups on Hardy Spaces for $$0

Journal of Fourier Analysis and Applications - In this paper, we discuss optimal constants and extremisers of Kato-smoothing estimates for the 2D Dirac equation. Smoothing estimates are...     

L ∞ andW 1, ∞ estimates for second order generalized difference methods on two-point boundary value problems

In this paper, we consider the second order generalized difference scheme for the two-point boundary value problem and obtain optimal order error estimates inL ^∞ andW ^{1, ∞}. The results in this paper perfect the theory of the second order generalized difference method.     

L P estimates for ∂-equation on generalized complex ellipsoids

The estimate of a holomorphic supporting function for the generalized complex ellipsoid in ?ⁿ is given, This domain is not decoupled. By using this estimate, the best possibleL ^p estimates for the ?-equation and some results of function theory on generalized complex ellipsoids are proved.     

Notes on Schneider’s stability estimates for convex sets

In 2009 Schneider obtained stability estimates in terms of the Banach–Mazur distance for several geometric inequalities for convex bodies in an n-dimensional normed space ${\mathbb{E}^n}$ . A unique feature of his approach is to express fundamental geometric quantities in terms of a single function ${\rho:\mathfrak{B} \times \mathfrak{B} \to \mathbb{R}}$ defined on the family of all convex bodies ${\mathfrak{B}}$ in ${\mathbb{E}^n}$ . In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric d _D on ${\mathfrak{B}}$ inducing, from our point of view, a finer topology than Banach–Mazur’s d _BM . Further, d _D induces a metric on the quotient ${\mathfrak{B}/{\rm Dil}^+}$ of ${\mathfrak{B}}$ by the relation of positive dilatation (homothety). Unlike its compact Banach–Mazur counterpart, d _D is only “boundedly compact,” in particular, complete and locally compact. The general linear group ${{\rm GL}(\mathbb{E}^n)}$ acts on ${\mathfrak{B}/{\rm Dil}^+}$ by isometries with respect to d _D , and the orbit space is naturally identified with the Banach–Mazur compactum ${\mathfrak{B}/{\rm Aff}}$ via the natural projection ${\pi:\mathfrak{B}/{\rm Dil}^+\to\mathfrak{B}/{\rm Aff}}$ , where Aff is the affine group of ${\mathbb{E}^n}$ . The metric d _D has the advantage that many geometric quantities are explicitly computable. We show that d _D provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of Schneider turn out to be valid with d _BM replaced by d _D .     

Degree estimates for C k‐piecewise polynomial subdivision surfaces

Piecewise polynomial subdivision surfaces are considered which consist of tri‐ or quadrilateral patches in a mostly regular arrangement with finitely many irregularities. A sharp estimate on the lowest possible degree of the polynomial patches is given. It depends on the smoothness and flexibility of the underlying subdivision scheme. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some estimates for maximal functions on Köthe function spaces

IfA is an invertiblen×n matrix with entries in the finite field F_q, letT _n (A) be its minimum period or exponent, i.e. its order as an element of the general linear group GL(n,q). The main result is, roughly, that $T_n (A) = q^{n - } (log n)^{2 + 0(1)} $ for almost everyA.     

Multilinear estimates for the Laplace spectral projectors on compact manifolds

The purpose of this Note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained by the authors (preprint: http://www.arxiv.org/abs/math/0308214) in dimension 2. We also give some related trilinear estimates. To cite this article: N. Burq et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Optimal Sobolev estimates for ¯∂ on convex domains of finite type

We prove that solutions for ¯ get 1/M-derivatives more than the data in L^p-Sobolev spaces on a bounded convex domain of finite type M by means of the integral kernel method. Also we prove that the Bergman projection is invariant under the L^p-Sobolev spaces of fractional orders by different methods from McNeal-Stein's. By using these results, we can get L^p-Sobolev estimates of order 1/M for the canonical solution for ¯. The author was supported by grant No. R01-2000-000-00001-0 from the Basic Research Program of the Korea Science&Engineering Foundation.     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Restrictions on local inhomogeneous Strichartz estimates for the Schrödinger equation

We find new necessary conditions for the estimate ${||u||_{L^{q}_{t} (\mathbb{R}; L^{r}_{x} (\mathbb{R}^{n}))} \lesssim\,||F||_{L^{{\tilde{q}}^{\prime}}_{t}(\mathbb{R};L^{{\tilde{r}}^{\prime}}_{x}(\mathbb{R}^{n}))}}$ , where u =  u(t, x) is the solution to the Cauchy problem associated with the free inhomogeneous Schrödinger equation with identically zero initial data and inhomogeneity F =  F(t, x).