FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2002, VOLUME 8, NUMBER 1, PAGES 141-150
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