Sobolev and Jawerth embeddings for spaces with variable smoothness and integrability. (English) Zbl 1184.46037
Summary: We consider the Triebel-Lizorkin spaces \(F^{s(\cdot)}_{p(\cdot),q(\cdot)}(\mathbb R^n)\) of variable smoothness and integrability introduced recently by L.Diening, P.Hästö and S.Roudenko in [Math.Z.262, No.3, 645–682 (2009; Zbl 1179.46029)]. Under certain regularity conditions on the function parameters involved, we show that
\[ F^{s_0(\cdot)}_{p_0(\cdot),q(\cdot)}(\mathbb R^n)\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q(\cdot)}(\mathbb R^n) \]
if
\[ s_0(x)\geq s_1(x)\text{ and }s_0(x)-\frac{n}{p_0(x)}=s_1(x)=\frac{n}{p_1(x)}\text{ for all }x\in\mathbb R^n \]
with embeddings of Sobolev and Bessel potential spaces included as special cases.
If \(\inf_{x\in\mathbb R^n}(s_0(x)-s_1(x))>0\), we recover also the analogue of the Jawerth embedding
\[ F^{s_0(\cdot)}_{p_0(\cdot),q_0(\cdot)}(\mathbb R^n)\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q_1(\cdot)}(\mathbb R^n) \]
for any \(q_0\), \(q_1\).
The proofs are based on the decomposition techniques of [loc. cit.] and work exclusively with the associated sequence spaces \(F^{s(\cdot)}_{p(\cdot),q(\cdot)}\).
\[ F^{s_0(\cdot)}_{p_0(\cdot),q(\cdot)}(\mathbb R^n)\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q(\cdot)}(\mathbb R^n) \]
if
\[ s_0(x)\geq s_1(x)\text{ and }s_0(x)-\frac{n}{p_0(x)}=s_1(x)=\frac{n}{p_1(x)}\text{ for all }x\in\mathbb R^n \]
with embeddings of Sobolev and Bessel potential spaces included as special cases.
If \(\inf_{x\in\mathbb R^n}(s_0(x)-s_1(x))>0\), we recover also the analogue of the Jawerth embedding
\[ F^{s_0(\cdot)}_{p_0(\cdot),q_0(\cdot)}(\mathbb R^n)\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q_1(\cdot)}(\mathbb R^n) \]
for any \(q_0\), \(q_1\).
The proofs are based on the decomposition techniques of [loc. cit.] and work exclusively with the associated sequence spaces \(F^{s(\cdot)}_{p(\cdot),q(\cdot)}\).
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E39 | Sobolev (and similar kinds of) spaces of functions of discrete variables |
