Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions
Tóm tắt
In this paper we analyse functions in Besov spaces
$$B^{1/q}_{q,\infty }(\mathbb {R}^N,\mathbb {R}^d),q\in (1,\infty )$$
, and functions in fractional Sobolev spaces
$$W^{r,q}(\mathbb {R}^N,\mathbb {R}^d),r\in (0,1),q\in [1,\infty )$$
. We prove for Besov functions
$$u\in B^{1/q}_{q,\infty }(\mathbb {R}^N,\mathbb {R}^d)$$
the summability of the difference between one-sided approximate limits in power q,
$$|u^+-u^-|^q$$
, along the jump set
$$\mathcal {J}_u$$
of u with respect to Hausdorff measure
$$\mathcal {H}^{N-1}$$
, and establish the best bound from above on the integral
$$\int _{\mathcal {J}_u}|u^+-u^-|^qd\mathcal {H}^{N-1}$$
in terms of Besov constants. We show for functions
$$u\in B^{1/q}_{q,\infty }(\mathbb {R}^N,\mathbb {R}^d),q\in (1,\infty )$$
that
0.1
$$\begin{aligned} \liminf \limits _{\varepsilon \rightarrow 0^+}\frac{1}{\varepsilon ^N}\int _{B_{\varepsilon }(x)} |u(z)-u_{B_{\varepsilon }(x)}|^qdz=0 \end{aligned}$$
for every x outside of a
$$\mathcal {H}^{N-1}$$
-sigma finite set. For fractional Sobolev functions
$$u\in W^{r,q}(\mathbb {R}^N,\mathbb {R}^d)$$
we prove that
0.2
$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+}\frac{1}{\varepsilon ^N}\int _{B_{\varepsilon }(x)}\frac{1}{\varepsilon ^N}\int _{B_{\varepsilon }(x)} |u\big (z\big )-u(y)|^qdzdy=0 \end{aligned}$$
for
$$\mathcal {H}^{N-rq}$$
a.e. x, where
$$q\in [1,\infty )$$
,
$$r\in (0,1)$$
and
$$rq\le N$$
. We prove for
$$u\in W^{1,q}(\mathbb {R}^N),1
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