Higher order fractional weighted homogeneous spaces: Characterization and finer embeddings

Abstract

In this article, for N >= 2, s is an element of (1, 2), p is an element of (1, N/s), sigma = s - 1and a is an element of [0, N-sp/2), we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi spaceD-a(s,p) (R-N) := <((CCR)-R-infinity(N))over bar>([center dot]s, p, a)where [u] s,p,a := (integral integral(RN x RN) |del u(x)- del u(y)|(p)/|x - y|(N+ sigma p) dx|x|(a) dy/|y|(a))(p),and higher order fractional weighted homogeneous space (W) over circle (s,p)(a) (R-N) := {u is an element of L-a(ps)*(R-N) :

del u

(Lap sigma*(RN)) + [u](s ,p,a) < infinity}with the weighted Lebesgue norm

u

(Lap sigma*(RN)) := integral(RN) |u(x)|(p)alpha*/|x|2ap alpha*/p dx) 1/p alpha*, where p alpha* = Np/N - alpha p for alpha = s, sigma. To achieve this, we prove that C-c(infinity)(R-N) is dense in (W) over circle (s,p)(a) (R-N) with respect to [center dot](s,p,a), and [center dot] (s,p,a) is an equivalent norm on (W) over circle (s,p)(a) (R-N). Further, we obtain a finer embedding of D-a(s,p)(R-N) into the Lorentz space L Np/N- sp-2a, p(R-N), where L Np/N- sp-2a, p(R-N) not subset of L-a(ps)*(R-N). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.