Let L := −Δ + V be a Schrödinger operator on\mathbb{R}^{n} with n\geqslant 3 and V\geqslant 0 satisfying \Delta ^{-1}V\in L^{\infty }(\mathbb{R}^{n}). Assume that φ: {R}^{n} × [0,∞) → [0,∞) is a function such that φ(x,,) is an Orlicz function, φ(•, t) \in A_{\infty }({R}^{n}) (the class of uniformly Muckenhoupt weights). Let w be an L-harmonic function on {R}^{n} with 0< C_{1}\leq w\leq C_{2}, where C_{1} and C_{2} are positive constants. In this article, the author proves that the mapping H_{\phi ,L} (\mathbb{R}^n ) \mathrel\backepsilon f \mapsto wf \in H_\phi (\mathbb{R}^n ) is an isomorphism from the Musielak-Orlicz-Hardy space associated with L,H_{\phi ,L} (\mathbb{R}^n ), to the Musielak-Orlicz-Hardy space H_\phi (\mathbb{R}^n ) under some assumptions on φ. As applications, the author further obtains the atomic and molecular characterizations of the space H_{\phi ,L} (\mathbb{R}^n ) associated with w, and proves that the operator {( - \Delta )^{ - 1/2}}{L^{1/2}} is an isomorphism of the spaces H_{\phi ,L} (\mathbb{R}^n ) and H_\phi (\mathbb{R}^n ). All these results are new even when φ(x, t) ≔ t^{p}, for all x \in \mathbb{R}^{n} and t \in [0,∞), with p ∞ (n/(n + μ_{0}), 1) and some μ_{0} \in (0, 1]., Sibei Yang., and Obsahuje seznam literatury
In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.
Let L1 = −Δ + V be a Schrödinger operator and let L2 = (−Δ)2 + V2 be a Schrödinger type operator on \mathbb{R}^{n}\left ( n\geqslant 5 \right ) where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s\geqslant n/2. The Hardy type space H_{L2}^{1} is defined in terms of the maximal function with respect to the semigroup \left\{ {{e^{ - t{L_2}}}} \right\} and it is identical to the Hardy space H_{L2}^{1} established by Dziubański and Zienkiewicz. In this article, we prove the Lp-boundedness of the commutator Rb = bRf - R(bf) generated by the Riesz transform R = {\nabla ^2}L_2^{ - 1/2} , where b \in BM{O_\theta }(\varrho ) , which is larger than the space BMO\left (\mathbb{R}^{n} \right ). Moreover, we prove that Rb is bounded from the Hardy space H_{L2}^{1} into weak L_{weak}^1 (\mathbb{R}^n )., Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang., and Obsahuje seznam literatury