Abstract
The present paper introduces a sharp Trudinger type inequality for harmonic functions based on the Cauchy-Riesz kernel function, which includes modified Poisson type kernel in a half plane considered by Xu et al. (Bound. Value Probl. 2013:262, 2013). As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded convex nonempty subsets of any Banach space, but also deduce the representation for set-valued maps and for scalar-valued maps of Dunford-Schwartz.
Highlights
The Trudinger inequality problem (TIP) is generated from the method of mathemati-D cal physics and nonlinear programming
Physicists have long been using so-called singular functions such as the Dirac delta function δ, T these cannot be properly defined within the framework of classical function theory
5 Conclusions In this paper, we obtained the representation of continuous linear maps in the set of all closed bounded convex nonempty subsets of any Banach space
Summary
The Trudinger inequality problem (TIP) is generated from the method of mathemati-. D cal physics and nonlinear programming. R p + q = n is the dimension of the Euclidean space Rn, the P = hypersurface was a hypercone with a singular point (the vertex) at the origin. They defined the generalized functions δ (k+ )(P) and δ (k+ )(P) as in the cases p, q < and p, q = , respectively. ). The hypersurface G = is a generalization of a hypercone P = with a singular point (the vertex) at the origin. Xn) = x + x + · · · + x p+ m – x p+ + · · · + x p+q m, R the G = hypersurface is a hypercone with a singular point (the vertex) at the origin.
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