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Methods Of Modern Mathematical Physics

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, where they are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motion to the analysis of partial differential equations and potential theory.

Probability theory - an analytic view, by Daniel W. Stroock. Pp 512. £30. 1994. ISBN 0-521-43123-9 (Cambridge University Press)

Moser—truedinger inequality on compact riemannian manifolds of dimension two

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev–Morrey space $H\mathcal{M}^{\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$
 with n∈ℕ and 1<q≤p<∞, which coincides with the usual critical Sobolev space $H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})$
 in the case of q=p. Indeed, we shall show the following interpolation inequality. If q<p, there exists a positive constant C
 p,q
 depending only on p and q such that GN
 $$ \|f\|_{{\mathcal{M}}_{r,\frac{q}{p}r}} \leq C_{p,q}r\|f \|_{{\mathcal{M}}_{p,q}}^{\frac{p}{r}}\bigl\|(-\Delta)^{\frac{n}{2p}} f\bigr\|_{{\mathcal{M}}_{p,q}}^{1-\frac{p}{r}} $$
 for all $u\in H\mathcal{M}^{\frac{n}{p}}_{p,q}( \mathbb{R}^{n})$
 and for all p≤r<∞. In the case of q=p, that is, the case of the critical Sobolev space $H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})$
 , the corresponding inequality was obtained in Ogawa (Nonlinear Anal. 14:765–769, 1990), Ogawa-Ozawa (J. Math. Anal. Appl. 155:531–540, 1991) and Ozawa (J. Func. Anal. 127:259–269, 1995) with the growth order $r^{1-\frac{1}{p}}$
 as r→∞. The inequality (GN) implies that the growth order as r→∞ is linear, which might look worse compared to the case of the critical Sobolev space. However, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brezis-Gallouet-Wainger type inequality in the critical Sobolev-Morrey space.

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

Hardy type inequalities are presented on balls with radius $R$ at the origin in ${\boldsymbol R}^n$ with $n=2$ at least. A special attention is paid on the behavior of functions on the boundary.

/pdf/hardy-type-inequalities-on-balls-3gahsemhv8.pdf

Hardy type inequalities on balls

In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert (9, 10) proved that the Hardy- Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary. We generalize their results by using the idea of Brezis and Nirenberg, and as a consequence, we shall prove the existence of pos- itive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.

/pdf/minimizing-problems-for-the-hardy-sobolev-type-inequality-2lgdmz1lej.pdf

Minimizing problems for the hardy-sobolev type inequality with the singularity on the boundary

We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.

Hidemitsu Wadade

Papers

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

Hardy type inequalities on balls

Minimizing problems for the hardy-sobolev type inequality with the singularity on the boundary

Remarks on the Rellich inequality