相关论文: Landau's converse to H\" older's inequality
Let $p>1$ and $1/p+1/q=1$. Consider H\"older's inequality $$ \|ab^*\|_1\le \|a\|_p\|b\|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra).…
We use the H\"{o}lder inequality for mixed exponents to prove some optimal variants of the generalized Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces with mixed exponents. Our results extend recent results of Araujo…
If $a,b$ are $n\times n$ matrices, Ando proved that Young's inequality is valid for their singular values: if $p>1$ and $1/p+1/q=1$, then $$ \lambda_k|ab^*|\le \lambda_k( \frac1p |a|^p+\frac 1q |b|^q ) \, \textit{ for all }k. $$ Later, this…
We show that Lieb's concavity theorem holds more generally for any unitary invariant matrix function $\phi:\mathbf{H}_+^n\rightarrow \mathbb{R}_+^n$ that is concave and satisfies H\"older's inequality. Concretely, we prove the joint…
Classical interpolation inequality of the type $\|u\|_{X}\leq C\|u\|_{Y}^{\theta}\|u\|_{Z}^{1-\theta}$ is well known in the case when $X$, $Y$, $Z$ are Lebesgue spaces. In this paper we show that this result may be extended by replacing…
A proof is given of Rosenthal's \(\ell_1\) theorem.
We consider the linearized Landau operator for which we provide simple proofs of hypoellipticity, and in particular we recover the recent results of H\' erau and Pravda-Starov \cite{herau-all}. Our arguments are elementary and in particular…
We give a new elementary proof of Landau's Prime Ideal Theorem. The proof is an extension of Richter's proof of the Prime Number Theorem. The main result contains other results related to the equidistribution of the prime ideal counting…
A new inequality, $(x)^{p}+(1-x)^{\frac{1}{p}}\leq1$ for $p \geq 1$ and $\frac{1}{2} \geq x \geq 0$ is found and proved. The inequality looks elegant as it integrates two number pairs ($x$ and $1-x$, $p$ and $\frac{1}{p}$) whose summation…
We prove an $\ell^p$-version of the coarse Baum-Connes conjecture for spaces that coarsely embedds into $\ell^q$-spaces for any $p$ and $q$ in $[1,\infty)$.
This work deals with the Landau equation in a bounded domain with the Maxwell reflection condition on the boundary for any (possibly smoothly position dependent) accommodation coefficient and for the full range of interaction potentials,…
We establish new optimal reversed Hardy-type inequalities on the cone of decreasing sequences from $\ell^p$-spaces with power weights, as well as estimates between different norms in Lorentz spaces of sequences. Based on these inequalities,…
Legendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left |\alpha-\frac{p}{q} \right | < \frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this…
In this note we prove a multilinear version of the reverse H\"older inequality in the theory of Muckenhoupt $A_p$ weights. We give two applications of this inequality to the study of multilinear weighted norm inequalities. First, we prove a…
Given a prime p and a positive integer m satisfying a certain inequality, the converse of Wolstenholme's Theorem is shown to hold for the product mp^k where k is any positive integer, generalizing a result by Helou and Terjanian.
We study the fractional $p$-Laplace equation $$ (-\Delta_p)^s u = 0 $$ for $0<s<1$ and in the subquadratic case $1<p<2$. We provide H\"older estimates with an explicit H\"older exponent. The inhomogeneous equation is also treated and there…
An approximation result for the bilinear Hilbert transform is proved and used for the inversion of the bilinear Hilbert transform. Also, p-Lebesgue points $(p\geq 1)$ are analyzed.
In this paper we prove a reverse H\"{o}lder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the…
The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov-Lawson for generalised Dirac operators as well as…
We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a…