Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint

• 作者: ZHU Maochun ZHENG Yifeng
• 来源期刊: 偏微分方程（英文版）
• 年，卷（期）: 2021,（2）
• 页码范围: 116-128
• 页数: 13页
• 分类号: O178
• 语种: 英文
• DOI: 10.4208/jpde.v34.n2.2

摘要

In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of R under the Lorentz-Sobolev norms constraint.For any 1＜q＜∞ and β≤(√π)q'≡ βq,q'=q/q-1,we obtain u∈(H)1/2,2(I),sup‖(-△)1/4u‖2,q≤1∫Ieβ|u(x)|q'dx≤c0|J|,and βq is optimal in the sense that u∈(H)1/2,2(I),sup‖(-△)1/4u‖2,q≤1∫Ieβ|u(x)|q'dx=+∞,for any β＞βq.Furthermore,when q is even,we obtain u∈(H)1/2,2(I),sup‖(-△)1/4u‖2,q≤1∫Ih(u)eβq|u(x)|q'dx=+∞,for any function h:[0,∞)→[0,∞)with limt→∞h(t)=∞.As for the key tools of proof,we use Green functions for fractional Laplace operators and the rearrangement of a convolution to the rearrangement of the convoluted functions.