Trudinger-Moser Type Inequality Under Lorentz-Sobolev Norms Constraint

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 infinity and beta <= (root pi)q' equivalent to beta q, q' = q/q-1, we obtain sup(u is an element of<(H)over tilde>1/2,2(I),parallel to(-Delta)1/4u parallel to 2,q) (<= 1) integral(e beta vertical bar mu(x)vertical bar q')(I) dx <= c(0)vertical bar I vertical bar, and beta(q) is optimal in the sense that sup(u is an element of(H) over tilde1/2,2(I),parallel to(-Delta)1/4u parallel to 2,q) (<= 1) integral(e beta vertical bar mu(x)vertical bar q')(I) dx=+infinity, for any beta > beta(q). Furthermore, when q is even, we obtain sup(u is an element of(H) over tilde1/2,2(I),parallel to(-Delta)1/4u parallel to 2,q) (<= 1) integral(h beta vertical bar mu(x)vertical bar q')(I) dx=+infinity, for any function h : [0, infinity) ->[0, infinity) with lim(t ->infinity)h(t)= infinity. 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.