Mathematical Sciences: Weighted norm inequalities and applications

我们将继续研究各种经典算子和与正交多项式展开相关的算子的加权范数不等式。这些不等式在微分方程中的应用也将被考察。要考虑的范数不等式涉及分数积分，Sobolev插值不等式，傅里叶变换，加权Hardy和Lebesgue积空间，n维Hardy不等式，Littlewood-Paley函数和Hilbert变换。在所有这些情况下的基本问题是理解加权范数不等式所适用的权函数的性质。与正交多项式展开相关的算子是Hermite和Laguerre多项式的Cesaro和算子和移植算子。弱型和强型范数不等式将通过得到精确的核估计来证明。微分方程的应用涉及退化椭圆方程解的正则性。在退化由两个权函数控制的情况下，问题包括格林函数的估计和与之相关的势的范数不等式。在权值相等的情况下，问题是当假设系数具有一定的平滑性时，权值如何影响解的平滑性。对于系数不光滑且在不同方向上退化不同的方程，也将研究哈纳克不等式的局部版本。

Work will continue on weighted norm inequalities for various classical operators and operators related to orthogonal polynomial expansions. Applications of these inequalities to differential equations will also be examined. The norm inequalities to be considered concern fractional integrals, Sobolev interpolation inequalities, Fourier transforms, weighted Hardy and Lebesgue product spaces, n-dimensional Hardy's inequality, Littlewood-Paley functions and Hilbert transforms. The basic problem in all these situations is that of understanding the character of the weight functions for which weighted norm inequalities hold. The operators related to orthogonal polynomial expansions are Cesaro sum operators and transplantation operators for Hermite and Laguerre polynomials. Weak and strong type norm inequalities will be proved by obtaining accurate kernel estimates. Applications to differential equations concern the regularity of solutions of degenerate elliptic equations. In case the degeneracy is governed by two weight functions, the problems include estimates of the Green function and norm inequalities for the potentials associated with it. With equal weights, the problem is how the weight affects the smoothness of the solutions when the coefficients are assumed to have some smoothness. A local version of Harnack's inequality will also be investigated for equations whose coefficients are not smooth and degenerate differently in different directions.