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Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity. (English) Zbl 1145.26010

Russ. Math. Surv. 60, No. 6, 1147-1164 (2005); translation from Usp. Mat. Nauk 60, No. 6, 139-156 (2005).
The author investigates multiplicative inequalities of the Gagliardo-Nirenberg type that connect partial moduli of continuity of functions with respect to different norms. For a function \(f\in L^p({\mathbb R}^n)\) (\(1\leq p<\infty\)) the modulus of continuity \(\omega(f;\delta)_p\) is defined as follows
\[ \omega(f;\delta)_p\equiv \sup_{| h| \leq \delta}\left( \int_{{\mathbb R}^n}| f(x)-f(x+h)| ^p\,dx\right)^{1/p}. \]
If the function \(f\) has a partial derivative \(D_jf\in L^p\) with respect to only a single variable \(x_j,\) the problem is to estimate the partial modulus of continuity
\[ \omega_j(f;\delta)_q\equiv \sup_{0\leq h\leq \delta}\left( \int_{{\mathbb R}^n}| f(x)-f(x+he_j)| ^q\,dx\right)^{1/q} \]
of the function with respect to the same variable in \(L^q,\) \(q>p.\) The author studies multiplicative inequalities for moduli of continuity in the scale of the Lorentz spaces \(L^{p,q}.\) Denote by \(\omega_j^r(f;\delta)_p\) the modulus of continuity of order \(r\) with respect to the variable \(x_j\) in \(L^p.\) In Theorem 1.2, he shows that
\[ \left(\int_0^{\infty}[h^{-\theta r} \omega_j^r(f;h)_p]^p\, {dh\over h}\right)^{1/p}\leq c| | f| | _{p_0}^{1-\theta}\|D_j^rf\|_{p_1}^{\theta}, \]
under the assumptions: \(1\leq p\leq \infty,\) \(1<p_1\leq \infty,\) \(0<\theta <1\) and \(1/p=(1-\theta)/p_0+\theta/p_1;\) moreover, \(f\in L^{p_0}({\mathbb R}^n)\) has the generalized derivative \(D_j^rf\in L^{p_1}({\mathbb R}^n)\) with respect to the variable \(j.\) This theorem fails to hold for \(p_1=1\) and \(p_0=\infty.\) Generally, the problems becomes much more complicated for \(p_1=1.\) In the paper, the author considers this case only for \(r=1.\) The following result is proved:
Theorem 1.4: Let \(1\leq p_0,s_0<\infty\) be given, and suppose that \(s_0=1\) if \(p_0=1.\) Let \(0<\theta<1\) and
\[ 1/p=(1-\theta)/p_0+\theta,\quad 1/s=(1-\theta)/s_0+\theta. \]
Assume that \(f\in L^{p_0,s_0}({\mathbb R}^n)\) has the generalized derivative \(D_jf\in L^1({\mathbb R}^n)\) with respect to the variable \(x_j.\) Then
\[ \left( \int_0^{\infty}[h^{-\theta}\omega_j(f;h)_{p,s}]^s\, {dh\over h}\right)^{1/s}\leq c\|f\|_{p_0,s_0}^{1-\theta} \|D_jf\|_1^{\theta}, \]
where \(c=c(p_0,s_0)[(1-\theta)\theta]^{-1/s}.\) From Theorem 1.4 one can derive other estimates for partial moduli of continuity and Besov norms. In particular, in Section 4, the author obtains some multiplicative inequalities involving norms in \(BV({\mathbb R}^n)\) of functions of bounded variations.
Reviewer: Rita Pini (Milano)

MSC:

26D99 Inequalities in real analysis
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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