变指数Besov和Triebel-Lizorkin空间的Franke-Jawerth嵌入
Franke-Jawerth embeddings for Besov and Triebel-Lizorkin spaces with variable exponents
摘要：
经典的Jawerth嵌入和Franke嵌入$$ F^{s_0}_{p_0,q}({\mathbb R}^n)\hookrightarrow B^{s_1}_{p_1,p_0}({\mathbb R}^n) \quad \mbox{and} \quad B^{s_0}_{p_0,p_1}({\mathbb R}^n)\hookrightarrow F^{s_1}_{p_1,q}({\mathbb R}^n) $$是Sobolev嵌入在Besov和Triebel-Lizorkin函数空间尺度之间的变形。我们证明了在Besov和Triebel-Lizorkin函数空间尺度上的Jawerth嵌入和Franke嵌入，如果$\inf_{x\in\mathbb{R}^n}(s_0(x)-s_1(x))>0$和$$ s_0(x) -\frac{n}{p_0(x)} = s_1(x) -\frac{n}{p_1(x)}, \quad x \in {\mathbb R}^n. $$，则它们的指数分别为$$ F^{s_0(\cdot)}_{p_0(\cdot),q(\cdot)}\hookrightarrow B^{s_1(\cdot)}_{p_1(\cdot),p_0(\cdot)} \quad \mbox{and} \quad B^{s_0(\cdot)}_{p_0(\cdot),p_1(\cdot)}\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q(\cdot)}, $$。我们只对相关序列空间$b^{s(\cdot)}_{p(\cdot),q(\cdot)}$和$f^{s(\cdot)}_{p(\cdot),q(\cdot)}$工作，这通过公知的分解技术来证明。我们还给出了一个不同的证明的Franke嵌入在常数指数的情况下，避免了对偶参数和插值。我们的结果也适用于2-微局部函数空间$B^{\mathbf{w}}_{p(\cdot),q(\cdot)}({\mathbb R}^n)$和$F^{\mathbf{w}}_{p(\cdot),q(\cdot)}({\mathbb R}^n)$，它们统一了变光滑空间和广义光滑空间的光滑尺度。
英文摘要：
The classical Jawerth and Franke embeddings $$ F^{s_0}_{p_0,q}({\mathbb R}^n)\hookrightarrow B^{s_1}_{p_1,p_0}({\mathbb R}^n) \quad \mbox{and} \quad B^{s_0}_{p_0,p_1}({\mathbb R}^n)\hookrightarrow F^{s_1}_{p_1,q}({\mathbb R}^n) $$ are versions of Sobolev embedding between the scales of Besov and Triebel-Lizorkin function spaces for $s_0>s_1$ and $$ s_0-\frac{n}{p_0} = s_1-\frac{n}{p_1}.$$ We prove Jawerth and Franke embeddings for the scales of Besov and Triebel-Lizorkin spaces with all exponents variable $$ F^{s_0(\cdot)}_{p_0(\cdot),q(\cdot)}\hookrightarrow B^{s_1(\cdot)}_{p_1(\cdot),p_0(\cdot)} \quad \mbox{and} \quad B^{s_0(\cdot)}_{p_0(\cdot),p_1(\cdot)}\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q(\cdot)}, $$ respectively, if $\inf_{x\in\mathbb{R}^n}(s_0(x)-s_1(x))>0$ and $$ s_0(x) -\frac{n}{p_0(x)} = s_1(x) -\frac{n}{p_1(x)}, \quad x \in {\mathbb R}^n. $$ We work exclusively with the associated sequence spaces $b^{s(\cdot)}_{p(\cdot),q(\cdot)}$ and $f^{s(\cdot)}_{p(\cdot),q(\cdot)}$, which is justified by well known decomposition techniques. We give also a different proof of the Franke embedding in the constant exponent case which avoids duality arguments and interpolation. Our results hold also for 2-microlocal function spaces $B^{\mathbf{w}}_{p(\cdot),q(\cdot)}({\mathbb R}^n)$ and $F^{\mathbf{w}}_{p(\cdot),q(\cdot)}({\mathbb R}^n)$ which unify the smoothness scales of spaces of variable smoothness and generalized smoothness spaces.
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