By means of the weight functions, the technique of real analysis and Hermite-Hadamard's inequality, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of logarithmic function and a best possible constant factor is given. Moreover, the equivalent forms, the operator expressions, the reverses and some particular cases are also considered.

In [LY] a differential Harnack inequality was proved for solutions to the heat equation on a Riemannian manifold. Inspired by this result, Hamilton first proved trace and matrix Harnack inequalities for the Ricci flow on compact surfaces [H0] and then vastly generalized his own result to all higher dimensions for complete solutions of the Ricci flow with nonnegative curvature operator [ H2]. So...

By means of weight functions and Hermite-Hadamard's inequality, and introducing a discrete interval variable, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of arc tangent function and a best possible constant factor is given, which is an extension of a published result. The equivalent forms and the operator expressions are also considered.

3 Some preparatory results 18 3.1 Green operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Harmonic functions and Harnack inequality . . . . . . . . . . . . . . . 22 3.3 Faber-Krahn inequality and mean exit time . . . . . . . . . . . . . . . 24 3.4 Estimates of the exit time . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∗Partially supported by SFB 701 of the Ge...

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace embeddings are derived.

We consider the Harnack inequality for harmonic functions with respect to three types of infinite-dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori mod...

Let Ω $\Omega$ be a bounded domain of R d $\mathbb {R}^d$ with Lipschitz boundary Γ $\Gamma$ . We define the Dirichlet-to-Neumann operator N $\mathcal {N}$ on L 2 ( ) $L_2(\Gamma )$ associated second-order elliptic A = − ∑ k , j 1 ∂ c l + b · 0 $A -\sum _{k,j=1}^d \partial _k (c_{kl} \, _l) \sum _{k=1}^d (b_k - _k(c_k \cdot )) a_0$ prove criterion for invariance closed convex set under action s...