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We study a space of potentials on the n-dimensional Euclidean space. They are constructed on the basis of rearrangement invariant spaces (RISs) by means of convolutions with kernels of general form. In particular, the spaces of classical potentials of Bessel and Riesz are included in the discussion. We establish the integral properties of potentials. For them the criteria for embedding in the RIS are found and explicit descriptions of the optimal RIS for these embeddings are obtained in case of the basic weighted Lorentz space. The aim of this paper is to give an explicit description of the optimal RIS X(Rn) for embeddings HEG(Rn)⊂X(Rn) in the case when the base RIS E(Rn) coincides with the weighted Lorentz space Λp(u).

It is shown that the well-known Trudinger–Moser inequality is equivalent to the embedding of the Sobolev space Wn1,∘ (Ω) into the Marcinkiewicz space M(ψ), and the function ψ is directly defined. It is shown that in the embedding theorem the Marcinkiewicz space M(ψn) can not be replaced even by the Lorentz space Λ(ψn) with the same fundamental function.

We study the multipliers of Fourier series on the Lorentz spaces, in particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Z in order to make it a multiplier of trigonometric Fourier series of space Lp,r [0; 1] in the Lq,r [0; 1]. In the paper there is a new multipliers theorem which is supplement of the well-known theorems, and given a counterexample.

Exact order estimates are obtained for the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes of periodic functions of many variables in the Lorentz space with the mixed norm.