Subcritical approach to sharp Hardy–Littlewood–Sobolev type inequalities on the upper half space

Abstract

Abstract. In this paper we establish the reversed sharp Hardy-LittlewoodSobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in [8]) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.

On a Conformally Invariant Integral Equation Involving Poisson Kernel

“…n−2 dy. By (7), for any |x ′ | < R − 1 we have |∇h i (R, x ′ )| ≤ Ch i (R, x ′ ) ≤ Cb p i for some C > 0 depending only on n and R. Therefore, subject to subsequence h i (R, x ′ ) → h(R, x ′ ) for some nonnegative function h ∈ C 1 (B R−1 ). Similar as in step 1, we split Pφ i as two parts Φ ′ i and Φ ′′ i with R replaced by R + 10.…”

Liouville theorems on the upper half space

“…(1.9) The sharp L p estimates were obtained in [8]. Later, more general extension operators on the upper half space were studied by Dou, Guo and Zhu [6] and Gluck [10].…”

“…They also provide us a new view point on how to obtain positive kernels for the extension operators. The elliptic properties and estimates, as well as the geometric applications of these extension operators were widely studied recently, see, for example, Caffarelli and Silvestre [3], Hang, Wang and Yang [11], Chen [5], Dou and Zhu [8], Dou, Guo and Zhu [6], Gluck [10], and references therein.…”

Subcritical approach to conformally invariant extension operators on the upper half space

“…These proofs all rely on the assumption that α + β ≥ 1. On the other hand, in [4] a subcritical approach was taken to prove an inequality of the form (1.2) for the kernel K α,1 and for the conformal exponents p = 2(n − 1)/(n + α − 2) and t = 2n/(n + α + 2) (which satisfy (1.4) with β = 1). Their approach also relies on the assumption that α + β ≥ 1.…”