We study the Hardy space of translated Dirichlet series $${\mathscr {H}}_{+}$$ . It consists on those $$\sum a_n n^{-s}$$ such that for some (equivalently, every) $$1 \le p < \infty $$ , translation {a_{n}}n^{-(s+\frac{1}{\sigma })}$$ belongs to {H}}^{p}$$ every $$\sigma >0$$ prove this set, endowed with topology induced by seminorms $$\left\{ \Vert \cdot _{2,k}\right\} _{k\in {\mathbb {N}}}$$ ...

For the isotropic stationary Lamé system with variable coefficients equipped with Dirichlet or surface stress boundary condition, we obtain a Carleman estimate such that (i) the right hand side is estimated in a weighted L2-space and (ii) the estimate includes nonhomogeneous surface displacement or surface stress. Using this estimate we establish the conditional stability in Sobolev’s norm of t...

with ai(x),pi(x) ∈ C1( ), pi(x) > 1,ai(x)≥ 0. Basing on the weighted variable exponent Sobolev space, a new kind of weak solutions of the equation is introduced. Whether the usual Dirichlet homogeneous boundary value condition can be imposed depends on whether ai(x) is degenerate on the boundary or not. If some of {ai(x)} are degenerate on the boundary, a partial boundary value condition is imp...

Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on Rn or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L)−1/2 on the Lp space. As an application, for 1 < p < 3+ ε, we establish the Lp boundedness...

The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincaré inequality but not the log-Sobolev one.

The Dirichlet distribution forms our first step toward understanding the DPM model. The Dirichlet distribution is a multi-parameter generalization of the Beta distribution and defines a distribution over distributions, i.e. the result of sampling a Dirichlet is a distribution on some discrete probability space. Let Θ = {θ1,θ2, . . . ,θn} be a probability distribution on the discrete space = { 1...