AbstractIn this work we analyze regularized optimal transport problems in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, the aim is to find a transport plan, which is another Radon measure on the product of the sets, that has these two measures as marginals and minimizes the sum of a certain linear cost function and a regularization term. We focus on regularization terms where a Young’s function applied to the (density of the) transport plan is integrated against a product measure. This forces the transport plan to belong to a certain Orlicz space. The predual problem is derived and proofs for strong duality and existence of primal solutions of the regularized problem are presented. Existence of (pre-)dual solutions is shown for the special case of $$L^p$$ L p -regularization for $$p\ge 2$$ p ≥ 2 . Moreover, two results regarding $$\varGamma $$ Γ -convergence are stated: The first is concerned with marginals that do not lie in the appropriate Orlicz space and guarantees $$\varGamma $$ Γ -convergence to the original Kantorovich problem, when smoothing the marginals. The second result gives convergence of a regularized and discretized problem to the unregularized, continuous problem.
AbstractWe investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.
We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$-spaces, where $1\leq p\leq \infty$. As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$-spaces is the zero operator.
Let $u$ and $\unicode[STIX]{x1D711}$ be two analytic functions on the unit disc $D$ such that $\unicode[STIX]{x1D711}(D)\subset D$ . A weighted composition operator $uC_{\unicode[STIX]{x1D711}}$ induced by $u$ and $\unicode[STIX]{x1D711}$ is defined by $uC_{\unicode[STIX]{x1D711}}f:=u\cdot f\circ \unicode[STIX]{x1D711}$ for every $f$ in $H^{p}$ , the Hardy space of $D$ . We investigate compactness of $uC_{\unicode[STIX]{x1D711}}$ on $H^{p}$ in terms of function-theoretic properties of $u$ and $\unicode[STIX]{x1D711}$ .
Let H be a Hilbert space with the usual product [x, y] and with an indefinite inner product (x, y) which, for some orthogonal decompositionin H, is defined bywhereand dim H1 = κ, a fixed positive integer.
Hydrophylita aquivalans was reared from the eggs of the damselfly, Ischnura verticalis; 92% of the eggs were parasitized. The behavior of the parasite is described within the host egg and during and after emergence.
Boundary value problems are formulated for the equation \[ ( ∗ ) L [ u ] = ∑ i , j = 1 n a i j ∂ 2 u ∂ x i ∂ x j + ∑ i = 1 n − 1 b i ∂ u ∂ x i + h x n ∂ u ∂ x n + c u = f ( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum \limits _{i = 1}^{n - 1} {{b_i}\frac {{\partial u}}{{\partial {x_i}}}} + \frac {h}{{{x_n}}}\frac {{\partial u}}{{\partial {x_n}}} + cu = f \] in a bounded domain G G in E n {E_n} with boundary ∂ G = S 1 ∪ S 2 \partial G = {S_1} \cup {S_2} where S 1 {S_1} is in x n = 0 {x_n} = 0 and S 2 {S_2} is in x n > 0 {x_n} > 0 . A uniqueness theorem is established for ( ∗ ) ( \ast ) when boundary data is only given on S 2 {S_2} for \[ h ( x 1 , ⋯ , x n − 1 , 0 ) ≧ 1 ; h({x_1}, \cdots ,{x_{n - 1}},0) \geqq 1; \] ; whereas an existence and uniqueness theorem for the Dirichlet problem is proved for h ( x 1 , x 2 , ⋯ , x n − 1 , 0 ) > 1 h({x_1},{x_2}, \cdots ,{x_{n - 1}},0) > 1 .