id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-846	Corcho, Adan J.; Mallqui, Lindolfo P.	L^2 Solutions for cubic NLS equation with higher order fractional elliptic/hyperbolic operators on R cross T and  R^2	2025	17	.pdf	application/pdf	6614	323	78	First note that S̃2 = 4 ⌊(C+K) 1 2α ⌋∑ n=⌊C 1 2α ⌋+1 √ C +K − n2α ≤ 4 √ K + ∫ ⌊(C+K) 1 2α ⌋ ⌊C 1 2α ⌋+1 √ C +K − z2αdz ≤ 4 √ K + ∫ (C+K) 1 2α C 1 2α √ C +K − z2αdz ≤ 4 √ K + √ K ( (C +K) 1 2α − C 1 2α ) . Considering the nonlinear change of variables ρ2α = z2α − C, and using that α > 1, C > 0 and K ≥ 1, we have J2 = K ∫ ∞ K 1 2α ρ2α−1 ρα ( ρ2α + C )1− 1 2α dρ ≤ K ∫ ∞ K 1 2α ρ−αdρ = 1 α− 1 KK 1−α 2α ≲α K. (2.17) Then, inserting (2.17) in (2.16) one obtains S− 2 ≲α K. Therefore, from (2.14) and the estimates obtained for S− 1 and S− 2 we have m ( G− 1,K ) ≲α K. By the same way we have m ( G− 2,K ) ≲α K. So, m ( G− K )	cache/ejde-846.pdf	txt/ejde-846.txt