On lower bound of the norm of integral convolution operator

E. D. Nursultanov ; K. S. Saidahmetov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 141-150

Résumé

We study the lower bound problem for the norm of integral convolution operator. We prove that if $1$, $K(x) \geq 0\ \forall x\in\mathbb R^n$ 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 $1$, 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 $$ C\sup_{e\in\mathfrak Q}\frac{1}{|e|^{1/p-1/q}} \biggl|\,\int_e K(x)\,dx\biggr|\leq\|A\|_{L_p\to L_q}. $$