The machinery of -dimensionally continuous functions is developed for the purpose of applying it to the Dirichlet problem for elliptic equations. With this extension of the space of continuous functions, new generalized definitions of classical and generalized solutions of the Dirichlet problem are given. Relations of these spaces of -dimensionally continuous functions to other known function spaces are studied. This has led to a new construction (seemingly more successful and closer to the classical one) of -dimensionally continuous functions, using which new properties of such spaces have been identified. The embeddings of the space in for and , and, in particular, in are proved. Previously, was shown to embed in , which secures the -dimensional continuity of generalized solutions. In the present paper, the more general embedding of in is verified and the corresponding exponents are shown to be sharp.

Bibliography: 33 titles.

开发了维连续函数的机制，目的是将其应用于椭圆方程的狄利克雷问题。通过连续函数空间的这种扩展，给出了狄利克雷问题的经典解和广义解的新广义定义。研究了这些维连续函数空间与其他已知函数空间的关系。这导致了维连续函数的新构造（似乎更成功，更接近经典），使用它已经确定了此类空间的新属性。证明了空间在for和 中的嵌入，特别是在 in中。之前，被证明嵌入，这确保了广义解的维连续性。在本文中，验证了更一般的in嵌入，并且显示出相应的指数是尖锐的。

参考书目：33 个标题。