The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations
Let \(X=B_p^{s,q}(\mathbb{R}^n)\) or \(X= F_p^{s,q}(\mathbb{R}^n)\) be the Besov and Triebel-Lizorkin space, respectively. The authors deal with the case \(0<p<1\) or \(0<q<1\), when these spaces are quasi-Banach. Let \(\widehat{X}\) denote the envelope of the quasi-Banach space \(X\). The authors show that \[ \widehat{B_p^{s,q}(\mathbb{R}^n)}=\widehat{F_p^{s,q}(\mathbb{R}^n)}=B_1^{s-n\left(\frac{1}{p}-1\right), 1}(\mathbb{R}^n) \] if both \(0<p<1\) and \(0<q<1\), \[ \widehat{B_p^{s,q}(\mathbb{R}^n)}=B_1^{s-n\left(\frac{1}{p}-1\right), q}\quad \text{and} \quad \widehat{F_p^{s,q}(\mathbb{R}^n)}=B_1^{s-n\left(\frac{1}{p}-1\right), 1}(\mathbb{R}^n), \] if \(0<p<1\) and \(1\leq q < \infty\), while \[ \widehat{B_p^{s,q}(\mathbb{R}^n)}=B_p^{s, 1}\quad \text{and} \quad \widehat{F_p^{s,q}(\mathbb{R}^n)}=F_p^{s, 1}(\mathbb{R}^n), \] when \(1\leq p < \infty\) and \(0<q<1\). Some applications of these results to PDEs in non-smooth domains are also given.
scientific article; zbMATH DE number 1539362
scientific article