The ball Banach fractional Sobolev inequality and its applications
Streszczenie
The authors obtain a fractional Sobolev inequality for Sobolev spaces $\dot {W}^{s,X}(\mathbb {R}^n)$ for ball Banach function spaces $X$ on $\mathbb {R}^n$ with the homogeneity and the non-collapse properties. More precisely, the authors show the existence of a positive constant $C$ such that, for any $f\in \dot {W}^{s,X}(\mathbb {R}^n) \cap X^{\frac {\alpha }{\alpha +s}}$, $$ \|f\|_{\dot {W}^{s,X}(\mathbb {R}^n)}\geq C\|f\|_{X^{\frac {\alpha }{\alpha +s}}}, $$ where $\alpha $ is the homogeneity index of $X$, $s\in (0,\min \,\{-\alpha ,1\})$, and $X^{\frac {\alpha }{\alpha +s}}$ is the $\frac {\alpha }{\alpha +s}$-convexification of $X$. Moreover, under some mild assumptions, the authors prove that the closure of $C_{\mathrm {c}}^\infty (\mathbb {R}^n)$ with respect to $\|\cdot \|_{\dot {W}^{s,X}(\mathbb {R}^n)}$ modulo constants is identified with $\dot {W}^{s,X}(\mathbb {R}^n)\cap X^{\frac {\alpha }{\alpha +s}}$. When $X$ is a Lebesgue space, these results reduce to the well-known Sobolev embeddings for which the restriction $s\in (0,\min \,\{-\alpha ,1\})$ is sharp. However, these results also provide new Sobolev embeddings for Morrey spaces, mixed-norm Lebesgue spaces, Lebesgue spaces with power weights, Besov–Triebel–Lizorkin–Bourgain–Morrey spaces, and Lorentz spaces. As in the case of the classical Sobolev inequality, these results have a wide range of applications.
