tailieunhanh - On the boundedness of integral operators

In this paper, we consider integal operators defined by kernel functions. It is well known the boundedness of such kind of operators as Shur Lemma type statements. But, the norm of operators was estimated by two integrals of kernel function. We obtain estimation of operators norm by one integral of kernel function. | Tr. J. of Mathematics 23 (1999) , 257 – 264. ¨ ITAK ˙ c TUB ON THE BOUNDEDNESS OF INTEGRAL OPERATORS ˙ Ekincio˘glu & I. ˙ A. Ikromov I. Abstract In this paper, we consider integal operators defined by kernel functions. It is well known the boundedness of such kind of operators as Shur Lemma type statements. But, the norm of operators was estimated by two integrals of kernel function. We obtain estimation of operators norm by one integral of kernel function. 1. Introduction Let Lp (Rn ) be the space of all integrable functions with degree p and denote by Lploc (Rn ) the space of all n by L∞ loc (R ) the space of p n local integrable functions with degree p in particular denote local bounded functions. Let D1 ⊂ Lp (Rn ) be a subspace of L (R ). Consider a linear operator T : D1 → D2 , where D2 ⊂ Lq (Rn ). Definition Operator T is called an operator of type (p, q) if there exists a real number c such that for any f ∈ D1 it holds the following inequality k T f kq ≤ c k f kp , where k · kp is a natural norm of the space Lp (Rn ) (see [3]). Note that if D1 is a dense set on Lp (Rn ) then the operator T has a bounded extension from Lp (Rn ) to Lq (Rn ). Let T be an integral operator with the kernel function K(x, y) and operator T is defined from C0∞ (Rn ) to L2loc (Rn ). It is well known the statements about bonudedness of integral operators. In the statements of R type Shur Lemma the L2 -norm of operators is estimated by both supy |K(x, y)|dx and R supx |K(x, y)|dy . . by two integrals of kernel functions (see [1]). 257 ˙ ˙ GLU, ˘ EKINC IO IKROMOV We consider the integral operator of type Z (T f)(y) = Rn K(x, y)f(x)dx . |y|n ∗ Denote by σA the set of Rn × Rn : σA ≡ {(x, y) : x ∈ Rn , y ∈ Rn : |y| ≤ A|X|}, where A is a fixed positive. Our main result consists of the following Theorem. Theorem . If supp K ⊂ σA , and there exists a number p > 2 such that Z |K( Rn y|x| , x)|pdy = ρ(x) ∈ L∞ (Rn ), |y|2 then the integral operator () has a bounded .