On Hardy–Steklov and geometric Steklov operators

Abstract

A new (non-Muckenhoupt type) weight characterization for the boundedness of the general Hardy-Steklov operator is obtained in the case 1 < p ≤ q < ∞. The estimates obtained for the norm of the Hardy-Steklov operator allow the limiting procedure and as a result the boundedness of the corresponding geometric Steklov operator is investigated.

Synchronization Analysis of Multiple Integral Inequalities Driven by Steklov Operator

“…Some new inequalities are demonstrated through the Hardy-Steklov operator (see [16][17][18][19][20]). For the sake of completeness, before asserting the key results, we recall the following definitions: Definition 1.…”

Bounded Operators on Mixed Norm Lebesgue Spaces

“…Derive from theorem 4.1 that K : L q1,q2 (R 2 , ν 1 , ν 2 ) → L p1,p2 (R 2 , µ 1 , µ 2 ) is bounded if the following two one-dimensional Hardy-Steklov type operators K 1 g(x 1 ) = b1(x1) a1(x1) g(y 1 )k 1 (x 1 , y 1 ) dy 1 and K 2 g(x 2 ) = b2(x1,x2) a2(x1,x2) g(y 2 )k 2 (x 1 , x 2 , y 2 ) dy 2 are bounded in corresponding Lebesgue spaces. Conditions for the boundedness of one-dimensional Hardy-Steklov type operator can be found in [12,13].…”

“…Finally, we write down sufficient conditions under which the two-dimensional Hardy-Steklov type operator g(y 2 )k 2 (x 1 , x 2 , y 2 ) dy 2 are bounded in corresponding Lebesgue spaces. Conditions for the boundedness of one-dimensional Hardy-Steklov type operator can be found in [12,13].…”

Some new integral inequalities via Steklov operator

“…

Hardy and Copson type inequalities have been studied by a large number of authors during the twentieth century and has motivated some important lines of study which are currently active. A large number of papers have been appeared involving Copson and Hardy inequalities (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] for more details).In this paper some Hardy-Steklov and Copson-Steklov type integral inequalities were established. Namely the integral inequalities were proved there.

…”