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On the embedding \(H^ w\subset V_ p\). (English) Zbl 0878.46017
For \(1\leq p<\infty\), let \(V_p\) be the set of functions \(f\) defined on an interval \([a,b]\) with finite \(p\)-variation \[ V_p(f;a,b)=\sup_G\Biggl\{\sum_{k=1}^N|f(s_k)- f(s_{k-1})|^p\Biggr\}^{1/p}, \] where the \(\sup\) is taken over all partitions \(a\leq s_0<s_1<\cdots< s_N\leq b\) of \([a,b]\). For a continuous nondecreasing function \(\omega(t)\) satisfying \(\omega(0)=0\) and \(\omega(t_1+ t_2)\leq\omega(t_1)+ \omega(t_2)\) let \(H^\omega\) be the set of continuous functions defined on \([0,1]\) whose modulus of continuity, defined by \(\sup\{|f(x+h)- f(x)|: 0<h<t, 0\leq x\leq 1-h\}\), is majorized by a constant times \(\omega(t)\). The author shows that the embedding \(H^\omega\subset V^p\) holds if and only if \(\omega(t)= O(t^{1/p})\) as \(t\to 0\).

MSC:
46E15 Banach spaces of continuous, differentiable or analytic functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A16 Lipschitz (Hölder) classes
42A20 Convergence and absolute convergence of Fourier and trigonometric series
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Keywords:
embedding
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References:
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