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 $1 < p \leq q < +\infty$, $K(x) \geq 0\ \forall x \in \mathbb{R}^n$ 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 $|e+e| \leq C \cdot |e|$, $|e|$ being the Lebesgue measure of the set $e$.

If $1<p<q<+\infty$, 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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Last modified: July 8, 2002