Peano curves and moduli of continuity.(English. Russian original)Zbl 0743.26008

Math. Notes 50, No. 2, 783-789 (1991); translation from Mat. Zametki 50, No. 2, 20-27 (1991).
The autor proves a Milne type inequality [see S. C. Milne, Adv. Math. 35, 129-157 (1980; Zbl 0449.26015)], namely for every non- increasing even function $$\varphi$$ on $$[0,\infty)$$ and every measurable a.e. finite function $$f$$ defined on the cube $$I^ n=[0,1]^ n$$ in $$\mathbb{R}^ n$$, and for any $$\delta_ i\in(0,1]$$ $$(i=1,\dots,n)$$ we have $\iint_{| s-t|\leq\delta_ 1\cdots\delta_ n/4^ n}\varphi(f^*(s)-f^*(t))ds dt\leq\iint_{| x_ i-y_ i|\leq\delta,i=1,\dots,n}\varphi(f(x)-f(y))dx dy.$ Here $$f^*$$ denotes the non-increasing rearrangement of $$f$$, $$f^*(t)=\inf\{s: m(\{x: | f(x)|>s\})\leq t\}$$. This gives a positive answer to a question posed by P. Oswald [Moduli of continuity of equimeasurable functions and approximation by algebraic polynomials in $$L^ p$$. Thesis, Odessa OGU (1978)]. As an application, the author proves that for every $$f$$ in $$L^ p(I^ n)$$ $$(0<p<\infty)$$, $$\omega_ p(f^*;\delta)\leq C_{p,n}\overline \omega_ p(f,\delta)$$ for $$0\leq\delta\leq 1$$, where $$\omega_ p$$ is the usual $$L^ 2$$-modulus of continuity and $$\overline\omega_ p(f,\delta)=\inf\{\omega_ p(f;\delta_ 1,\dots,\delta_ n); \delta _ 1\dots\delta_ n=\delta$$, $$0\leq\delta_ i\leq 1\}$$. The case of $$1\leq p<\infty$$ was earlier proved by Oswald [loc. cit.].

MSC:

 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26D15 Inequalities for sums, series and integrals

Zbl 0449.26015
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References:

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