Embeddings in Grand Variable Exponent Function Spaces

New scales of grand variable exponent Hajlasz–Sobolev and Holder spaces are introduced. Embeddings between these spaces are established under the log-Holder continuity condition on exponent functions of spaces. Spaces are defined, generally speaking, on quasi-metric measure spaces with doubling condition (spaces of homogeneous type) but the results are new even for Euclidean domains with Lebesgue measure. Sobolev embeddings for domains with Lipschitz boundaries in $${\mathbb {R}}^n$$ are also derived in the framework of new scales of grand variable exponent Lebesgue spaces. The proof of the latter result is based on the appropriate estimates for the sharp maximal function which are consequence of the sharp variant of the Rubio de Franca’s extrapolation result for variable exponent Lebesgue spaces. To prove the main results of this paper, we establish sharp bounds for norms of appropriate function spaces. Some essential properties of grand variable exponent Hajlasz–Sobolev and Holder spaces are investigated as well.