## Maximal inequality in $$(s,m)$$-uniform domains.(English)Zbl 1026.46022

The author deals with Sobolev spaces on bounded domains in $$\mathbb{R}^n$$. A new class of bounded domains is defined. The domains are called $$(s,m)$$-uniform, $$s\geq 1$$, $$0<m\leq 1$$, and they form a subclass of $$s$$-John domains. If a domain is bounded and its boundary is locally a graph of a $$\lambda$$-Hölder function, $$0<\lambda\leq 1$$, then it is $$(1/\lambda,\lambda)$$-uniform. The main theorem asserts that if $$\Omega\subset \mathbb{R}^n$$ is $$(s,m)$$-uniform, $$1\leq s<n/(n-1)$$ and $$0<m\leq 1$$, then every function $$u$$ belonging to the Sobolev space $$W^{1,p}(\Omega)$$ satisfies the inequality $|u(x)-u(y)|\leq C|x-y|^\alpha \big(\mathcal M\nabla u(x)+\mathcal M\nabla u(y)\big) \qquad x,y\in \Omega,\quad \alpha=\frac ms (n-s(n-1)) .$ Here $$\mathcal M$$ denotes the Hardy-Littlewood maximal function. The inequality was previously considered by P. Hajłasz and O. Martio [J. Funct. Anal. 143, 221-246 (1997; Zbl 0877.46025)]. The result extends the earlier estimates to a wider class of domains. The paper is based on techniques developed by its forerunners.

### MSC:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

### Keywords:

$$(s,m)$$-uniform domains; $$s$$-John domains

Zbl 0877.46025
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