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Gaussian estimates for Schrödinger perturbations

Krzysztof Bogdan, Karol Szczypkowski (2014)

Studia Mathematica

We propose a new general method of estimating Schrödinger perturbations of transition densities using an auxiliary transition density as a majorant of the perturbation series. We present applications to Gaussian bounds by proving an optimal inequality involving four Gaussian kernels, which we call the 4G Theorem. The applications come with honest control of constants in estimates of Schrödinger perturbations of Gaussian-type heat kernels and also allow for specific non-Kato perturbations.

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl, David E. Edmunds (2003)

Studia Mathematica

For a precompact subset K of a Hilbert space we prove the following inequalities: n 1 / 2 c ( c o v ( K ) ) c K ( 1 + k = 1 k - 1 / 2 e k ( K ) ) , n ∈ ℕ, and k 1 / 2 c k + n ( c o v ( K ) ) c [ l o g 1 / 2 ( n + 1 ) ε ( K ) + j = n + 1 ε j ( K ) / ( j l o g 1 / 2 ( j + 1 ) ) ] , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and ε k ( K ) and e k ( K ) denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly...

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