Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

8902952 惠登 要进行的调查的核心是 发展的积分不等式称为加权范数 不等式 不等式，比较加权 一个给定的函数的范数与它的图像在一些 变换（可能使用不同的权重）是基本的 在微分方程的分析中。 解 方程通常用积分变换表示， 涉及奇异核。 如果能建立基本的， 相对于变换的加权不等式，则 可以证明，解决方案可以在给定的范围内找到。 类的函数。 分析涉及分数积分，Sobolev和 Poincare不等式，插值不等式，Poisson 积分和傅立叶变换。 退化椭圆和 抛物方程，包括向量场以外的 将研究梯度场。 此外，工作将 为积极的全球存在寻求条件 一类半线性退化椭圆方程的解 方程 还计划开展工作，以确定 经典不等式可以推广到齐次不等式。 空间. 重点将放在是否 一阶向量场的Sobolev和Poincare不等式 （除了普通梯度）可以结转到这些 空间.

8902952 Wheeden Central to the investigations to be carried out is the development of integral inequalities known as weighted norm inequalities. The inequalities, which compare the weighted norm of a given function with that of its image under some transform (using possibly a different weight) are fundamental in the analysis of differential equations. Solutions of the equations are expressed in terms of integral transforms, often involving singular kernels. If one can establish basic, weighted inequalities relative to the transforms, it is then possible to show that solutions can be found within given classes of functions. The analysis involves fractional integrals, Sobolev and Poincare inequalities, interpolation inequalities, Poisson integrals and Fourier transforms. Degenerate elliptic and parabolic equations, including vector fields other than the gradient fields will be studied. In addition, work will be done seeking conditions for the global existence of positive solutions for a class of semilinear degenerate elliptic equations. Work is also planned in determining the extent to which classical inequalities can be carried over to homogeneous spaces. Emphasis will be placed on the question of whether the Sobolev and Poincare inequalities for first order vector fields (other than the ordinary gradient) can be carried over to these spaces.