FPM 2002, vol. 8, no. 1, pp. 141-150

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 $L$ _p to $L$ _q, 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 $L$ _p to $L$ _q, 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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