(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

View as HTML     View as gif image    View as LaTeX source

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

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

is a bounded operator from $L$p to $L$q, then there exists a constant $C\left(p,q,n\right)$ such that

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

Here $Q\left(C\right)$ 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 $C\left(p,q,n\right)$ such that

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

All articles are published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k02/k021/k02112h.htm