Embeddings of Sobolev-type spaces into generalized Hölder spaces involving $$k$$ -modulus of smoothness
Tóm tắt
We use an estimate of the
$$k$$
-modulus of smoothness of a function
$$f$$
such that the norm of its distributional gradient
$$|\nabla ^kf|$$
belongs locally to the Lorentz space
$$L^{n/k, 1}({\mathbb {R}}^n),\,k \in {\mathbb {N}},\,k\le n$$
, and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces. These spaces are modelled upon rearrangement-invariant Banach function spaces
$$X({\mathbb {R}}^n)$$
. Target spaces of our embeddings are generalized Hölder spaces defined by means of the
$$k$$
-modulus of smoothness
$$(k\in {\mathbb {N}})$$
. General results are illustrated with examples.
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