FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 1, PAGES 141-150

On lower bound of the norm of integral convolution operator

E. D. Nursultanov
K. S. Saidahmetov

Abstract

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We study the lower bound problem for the norm of integral convolution operator. We prove that if 1 < p £ q < +¥, $ K(x) \geq 0\ \forall x\in \mathbb{R}^n $ and the operator

$$ (Af)(x) = \int_{\mathbb{R}^n} K(x-y) f(y) dy = K*f $$

is a bounded operator from Lp to Lq, then there exists a constant C(p,q,n) such that

$$ C \sup_{e \in Q(C)} \frac{1}{|e|^{1/p-1/q}} \int_e K(x) dx \leq \|A\|_{L_p\to L_q}. $$

Here Q(C) is the set of all Lebesgue measurable sets of finite measure that satisfy the condition |e+e| £ C × |e|, |e| being the Lebesgue measure of the set e.

If 1 < p < q < + ¥, the operator A is a bounded operator from Lp to Lq, and $ \mathfrak Q $ is the set of all harmonic segments, then there exists a constant C(p,q,n) such that

$$ C \sup_{e \in \mathfrak Q} \frac{1}{|e|^{1/p-1/q}} \biggl| \int_e K(x) dx \biggr| \leq \|A\|_{L_p \to L_q}. $$

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