Let ΔSn denote the Laplace-Beltrami operator on n-dimensional unit sphere Sn, n≥2. In this paper we show that‖eitΔSnf‖L4([0,2π)×Sn)≤C‖f‖Wα,4(Sn) holds if α>(n−2)/4. The range of α is sharp in sense that estimate fails for α<(n−2)/4. As a consequence, obtain space-time Lp-estimates eitΔSn 2≤p≤∞. We also prove maximal f→sup0≤t<2π⁡|eitΔSnf| bounded from Wα,2(Sn) to L6n/(3n−2)(Sn) α>1/3 whenever f ...

In this short note, by using a general three circles theorem, we show the rigidity of sharp Bezout estimate first found Gang Liu on nonnegatively curved Riemann surfaces.

A one-parameter family of smooth hypersurfaces {Mt} ⊂ R flows by mean curvature if zt = H(z) = ∆Mtz , (0.1) where z are coordinates on R and H = −Hn is the mean curvature vector. In this note, we prove sharp gradient and area estimates for graphs flowing by mean curvature. Thus, each Mt is assumed to be the graph of a function u(·, t). So, if z = (x, y) with x ∈ R, then Mt is given by y = u(x, ...

The wave equation, ∂ttu = ∆u, in R, considered with initial data u(x, 0) = f ∈ H(R) and u′(x, 0) = 0, has a solution which we denote by 1 2 (e √ −∆f + e−it √ −∆f). We give almost sharp conditions under which sup0<t<1 |e ±it √ −∆f | and supt∈R |e ±it √ −∆f | are bounded from H(R) to L(R).

Let S stand for the usual class of univalent regular functions in the unit disk U = {z : |z| < 1} normalized by f(z) = z + a2z + . . . in U , and let S be its subclass defined by restricting |f(z)| < M in U , M ≥ 1. We consider two classical problems: Bombieri’s coefficient problem for the class S and the sharp estimate of the fourth coefficient of a function from S . Using Löwner’s parametric ...

The class of Bipartite Distance Hereditary (BDH) graphs is the intersection between bipartite domino-free and chordal bipartite graphs. Graphs in both the latter classes have linearly many maximal bicliques, implying the existence of polynomial-time algorithms for computing the associated Galois lattice. Such a lattice can indeed be built in O(m×n) worst case-time for a domino-free graph with m...

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.