Muhammad ASIF, Usman ASHRAF, Alexander MESKHI

Kernel operators on the upper half-space: boundedness and compactness criteria

We establish necessary and sufficient conditions on a weight v governing the trace inequality |\hat{K}f|Lqv(\hat{E}) \leq C|f|Lp(E), where E is a cone on a homogeneous group, \hat{E}: = E \times R+ and \hat{K} is a positive kernel operator defined on \hat{E}. Compactness criteria for this operator are also established.

Anahtar Kelimeler:

Operators with positive kernels, upper half-space, potentials, homogeneous groups, trace inequality, boundedness, compactness, weights

Kernel operators on the upper half-space: boundedness and compactness criteria

Keywords:

Operators with positive kernels, upper half-space, potentials, homogeneous groups, trace inequality, boundedness, compactness, weights,

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Adams, D.R.: A trace inequality for generalized potentials. Studia Math. 48, 99–105 (1973).
Ando, T.: On the compactness of integral operators. Indag. Math. (N.S.), 24, 235–239 (1962).
Ashraf, U., Asif, M., Meskhi, A.: Boundedness and compactness of positive integral operators on cones of homogeneous groups. Positivity 13, 497–518 (2009).
Bloom, S., Kerman, R., Weighted norm inequalities for operators of Hardy type. Proc. Amer. Math. Soc. 113, 135–141 (1991).
Edmunds, D.E., Kokilashvili, V., Meskhi, A.: Bounded and Compact Integral Operators. Dordrecht. Kluwer Academic Publishers 2002.
Folland, G.B., Stien, E.M.: Hardy Spaces on Homogeneous Groups. Princeton. Princeton University Press 1987.
Gabidzashvili, M., Genebashvili, I., Kokilashvili, V. Two-weight inequalities for generalized potentials. Trudy Mat. Inst. Steklov 194, 89–96 (1992) (in Russian). English transl. Proc. Steklov Inst. Math. 94, 91–99 (1993).
Genebashvili, I.: Carleson measures and potentials defined on the spaces of homogeneous type. Bull. Georgian Acad. Sci. 135, 505–508 (1989).
Jain, P., Jain, P.K., Gupta, B.: Higher dimensional compactness of Hardy operators involving Oinarov-type kernels. Math. Ineq. Appl. 9, 739–748 (2002).
Kantorovich, L.P., Akilov, G.P.: Functional Analysis. Oxford. Pergamon 1982.
Kokilashvili, V., Krbec, M.: Weighted Inequalities in Lorentz and Orlicz Spaces. Singapore. World Scientific 1991. Kokilashvili, V., Meskhi, A.: Boundedness and compactness criteria for the generalized truncated potentials. Proc. Steklov Math. Inst. RAN 232, 164–178 (2001) (in Russian). Engl. Transl. Proc. Steklov Inst. Math. 232 (2001).
Kokilashvili, V., Meskhi, A., Persson, L.E.: Weighted Norm Inequalities for Integral Transforms with Product Kernels. New York. Nova Science Publishers 2009.
Krasnoselski´ı, M.A., Zabreiko, P.P., Pustilnik, E.I., Sobolevski´ı, P.E.: Integral Operators in Spaces of Summable Functions. Moscow. Nauka 1966 (in Russian). Engl. Transl. Noordhoff International Publishing 1976.
Kufner, A., Persson, L.E.: Integral Inequalities with Weights. Prague. Academy of Sciences of the Czech Republic 2000.
Luor, D.H.: On the equivalence of weighted inequalities for a class of operators. Proc. Roy. Soc. Edinburgh 141 A, 1071–1081 (2011).
Maz’ya, V.G.: Sobolev Spaces, Berlin. Springer 1985.
Meskhi, A.: Solution of some weight problems for the Riemann-Liouville and Weyl operators. Georgian Math. J. 5, 565–574 (1998).
Meskhi, A.: Criteria of the boundedness and compactness for generalized Riemann-Lioville operators. Real Anal. Exchange 26, 217–236 (2000/2001).
Meskhi, A.: Criteria for the boundedness and compactness of operators with positive kernels. Proc. Edinburgh Math. Soc. 44, 267–284 (2001).
Meskhi, A.: Measure of Non-compactness for Integral Operators in Weighted Lebesgue Spaces. New York. Nova Science Publishers 2009.
Oinarov, R.: Two-sided estimate of certain classes of integral operators. Tr. Math. Inst. Steklova 204, 240–250 (1993) (in Russian).
Opic, B., Kufner, A.: Hardy-type Inequalities. Pitman Research Notes in Math. Series 219, Longman Sci. and Tech. 19 Prokhorov, D.V.: On the boundedness of a class of integral operators. J. London Math. Soc. 61, 617–628 (2000). Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. London. Gordon and Breach Science Publishers 1993.
Sawyer, E.T, Wheeden, R.L, Zhao, S.: Weighted norm inequalities for operators of potential type and fractional maximal functions. Potential Anal. 5, 523–580 (1996).
Sinnamon, G.: One-dimensional Hardy-inequalities in many dimensions. Proc. Royal Soc. Edinburgh A 128, 833– 848 (1998).
Sinnamon, G., Stepanov, V.: The weighted Hardy inequality: new proof and the case p = 1 . J. London Math. Soc. 54, 89–101 (1996).
Wedestig, A.: Weighted inequalities of Hardy-type and their limiting inequalities. PhD, Lulea University of Technology, Department of Mathematics, 2003.
Wheeden, R.L., Wilson, J.M.: Weighted norm estimates for gradients of halfspace extensions. Indiana Univ. Math. J. 44, 917–969 (1995).