2016/11/28 by Helena F. Gonçalves, Gonçalves, Helena F., Henning Kempka +3
Mathematics · Medicine · #Advanced Mathematical Physics Problems #Advanced Harmonic Analysis Research #Soft tissue tumor case studies
paper · pdf · doi:10.48550/arxiv.1611.08985
The classical Jawerth and Franke embeddings Fs0p0,q(
mathbb\nRn)
hookrightarrow Bs1p1,p0(
mathbb Rn)
quad
mboxand
quad\nBs0p0,p1(
mathbb Rn)
hookrightarrow Fs1p1,q(
mathbb Rn)\n are versions of Sobolev embedding between the scales of Besov and\nTriebel-Lizorkin function spaces for s0>s1 and s0-
fracnp0 =\ns1-
fracnp1. We prove Jawerth and Franke embeddings for the scales of\nBesov and Triebel-Lizorkin spaces with all exponents variable \nFs0(
cdot)p0(
cdot),q(
cdot)
hookrightarrow\nBs1(
cdot)p1(
cdot),p0(
cdot)
quad
mboxand
quad\nBs0(
cdot)p0(
cdot),p1(
cdot)
hookrightarrow\nFs1(
cdot)p1(
cdot),q(
cdot), respectively, if\n\infx\∈\ℝn(s0(x)-s1(x))>0 and s0(x) -
fracnp0(x) =\ns1(x) -
fracnp1(x),
quad x
in
mathbb Rn. We work exclusively\nwith the associated sequence spaces bs(\⋅)p(\⋅),q(\⋅) and\nfs(\⋅)p(\⋅),q(\⋅), which is justified by well known\ndecomposition techniques. We give also a different proof of the Franke\nembedding in the constant exponent case which avoids duality arguments and\ninterpolation.\n Our results hold also for 2-microlocal function spaces\nB\wp(\⋅),q(\⋅)( mathbb Rn) and\nF\wp(\⋅),q(\⋅)( mathbb Rn) which unify the smoothness\nscales of spaces of variable smoothness and generalized smoothness spaces.\n