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We prove embeddings of Sobolev and Hardy–Sobolev spaces into Besov spaces built upon certain mixed norms. This gives an improvement of the known embeddings into usual Besov spaces. Applying these results, we obtain Oberlin-type estimates of Fourier transforms for functions in Sobolev spaces \(W_1^1(\mathbb {R}^n).\)

Embedding theorems for Sobolev and Hardy–Sobolev spaces and estimates of Fourier transforms