New scales of grand variable exponent Hajłasz–Sobolev and Hölder spaces are introduced. Embeddings between these spaces are established under the log-Hölder 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 Francá’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 Hajłasz–Sobolev and Hölder spaces are investigated as well.

引入了大变量指数 Hajłasz-Sobolev 和 Hölder 空间的新尺度。这些空间之间的嵌入是在空间指数函数的 log-Hölder 连续性条件下建立的。一般来说，空间是在具有加倍条件的拟度量测度空间（齐次类型空间）上定义的，但即使对于具有 Lebesgue 测度的欧几里得域，结果也是新的。在\({\mathbb {R}}^n\) 中具有 Lipschitz 边界的域的 Sobolev 嵌入也在大变量指数 Lebesgue 空间的新尺度的框架中导出。后一个结果的证明基于对尖锐极大函数的适当估计，该函数是 Rubio de Francá 对可变指数 Lebesgue 空间的外推结果的尖锐变体的结果。为了证明本文的主要结果，我们为适当的函数空间的范数建立了严格的界限。还研究了大变量指数 Hajłasz-Sobolev 和 Hölder 空间的一些基本性质。