@Article{JPDE-34-116, author = {Zhu , Maochun and Zheng , Yifeng}, title = {Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint}, journal = {Journal of Partial Differential Equations}, year = {2021}, volume = {34}, number = {2}, pages = {116--128}, abstract = {

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 $β≤ \big(\sqrt{π} \big)^{q'} \equiv β_q, q'= \frac{q}{q-1}$, we obtain

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and $β_q$ is optimal in the sense that

image.png

for any $β>β_q$. Furthermore, when $q$ is even, we obtain

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for any function $h : [0,∞)→[0,∞)$ with lim$_{t→∞} h(t) = ∞$. 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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v34.n2.2}, url = {http://global-sci.org/intro/article_detail/jpde/19183.html} }