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Volume 34, Issue 2
Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint

Maochun Zhu & Yifeng Zheng

DOI: 10.4208/jpde.v34.n2.2

J. Part. Diff. Eq., 34 (2021), pp. 116-128.

Published online: 2021-05

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  • 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

image.png

and $β_q$ is optimal in the sense that

image.png

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

image.png

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.

  • Keywords

Trudinger-Moser inequality, Lorentz-Sobolev space, bounded intervals.

  • AMS Subject Headings

46E30, 46E35, 26D15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhumaochun2006@126.com (Maochun Zhu)

2590519973@qq.com (Yifeng Zheng)

  • BibTex
  • RIS
  • TXT
@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

image.png

and $β_q$ is optimal in the sense that

image.png

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

image.png

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} }
TY - JOUR T1 - Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint AU - Zhu , Maochun AU - Zheng , Yifeng JO - Journal of Partial Differential Equations VL - 2 SP - 116 EP - 128 PY - 2021 DA - 2021/05 SN - 34 DO - http://doi.org/10.4208/jpde.v34.n2.2 UR - https://global-sci.org/intro/article_detail/jpde/19183.html KW - Trudinger-Moser inequality, Lorentz-Sobolev space, bounded intervals. AB -

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

image.png

and $β_q$ is optimal in the sense that

image.png

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

image.png

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.

MaochunZhu & YifengZheng. (2021). Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint. Journal of Partial Differential Equations. 34 (2). 116-128. doi:10.4208/jpde.v34.n2.2
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