Note that in [6] the above results were generalized to the case of considerably more general operator ideals.
Moreover, she treat, a more general operator that includes, as a particular case, the -laplacian operator.
However, in this paper we will be mainly interested in studying a more general operator equation of the form begin{aligned} Lx=Nx, end{aligned} (1.1) where L is not necessarily invertible.
However, for various applications it is also important to consider the mapping properties of the more general operator (H_{k}(f)(x):=int_{0}^{x}k x,t f(t),dt), where (k x,t)) is a kernel (a non-negative and measurable function on ({(x,t):0leq x
The main contribution of this paper lies in generalizing this operator in three different ways: to also include matrix-valued potential functions as in Eq. (1.5), to the more general operator in Eq. (1.4), and lastly to settings where the operator acts on functions defined on subsets of the real line.
Regarding specifically the ( p, q ) -Laplacian ( p ≠ q ), Figueiredo proved, as a particular case of his main result in [1] (which was obtained for a problem involving a more general operator), the existence of a nontrivial weak solution for the following problem: { − Δ p u − Δ q u + | u | p − 2 u + | u | q − 2 u = λ f ( u ) + | u | p ⋆ − 2 u in R N, u ≥ 0, (5).
This article aims at generalizing the setting of Lieb Thirring inequalities to more general operators than the magnetic or non-magnetic Schrödinger operators.
The construction is also extended to tuples of more general operators allowing smooth functional calculii.
More general boundary conditions or more general operators ϕ were considered in [4 6, 9, 12].
18 below; extensions to more general operators have been given in [50, 53].
Analysis becomes much more complicated for more general operators even if we assume that the inner propagation is simple.
