Reverses of the Triangle Inequality in Banach Spaces

New Reverses of Schwarz, Triangle and Bessel Inequalities in Inner Product Spaces

New reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces are pointed out. These results complement the recent ones obtained by the author in an earlier paper. Further, they are employed to establish new Gruss type inequalities. Finally, some natural integral inequalities are stated as well.

Reverse Triangle Inequalities for Riesz Potentials and Connections with Polarization

We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.

On the Boundedness of The Bilinear Hilbert Transform along "non-flat" smooth curves. The Banach triangle case ($L^r,\: 1\leq r<\infty$)

We show that the bilinear Hilbert transform $H_Γ$ along curves $Γ=(t,-γ(t))$ with $γ\in\mathcal{N}\mathcal{F}^{C}$ is bounded from $L^{p}(\mathbb{R})\times L^{q}(\mathbb{R})\,\rightarrow\,L^{r}(\mathbb{R})$ where $p,\,q,\,r$ are Hölder indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p<\infty$, $1<q\leq\infty$ and $1\leq r<\infty$. Here $\mathcal{N}\mathcal{F}^{C}$ stands for a wide class of smooth "non-flat" curves near zero and infinity whose precise definition is given in Section 2. This continues author's earlier work on this topic, extending the boundedness range of $H_Γ$ to a…

Characterization of a generalized triangle inequality in normed spaces

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{μ_1}+...+\frac{|x_n|^p}{μ_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

On Some Discrete Inequalities in Normed Linear Spaces

Some sharp discrete inequalities in normed linear spaces are obtained. New reverses of the generalised triangle inequality are also given.

Problems with mixed boundary conditions in Banach spaces

Using Leray-Schauder degree or degree for $α$-condensing maps we obtain the existence of at least one solution for the boundary value problem of the type \[ \left\{\begin{array}{lll} (φ(u' ))' = f(t,u,u') & & \\ u(T)=0=u'(0), & & \quad \quad \end{array}\right. \] where $φ: X\rightarrow X $ is a homeomorphism with reverse Lipschitz such that $φ(0)=0$, $f:\left[0, T\right]\times X \times X \rightarrow X $ is a continuous function, $T$ a positive real number and $X$ is a real Banach space.

Maz'ya-Shaposhnikova表示法用于具有弱反倍增性质的同质类型空间上的球准Banach函数空间的准范数

Maz'ya--Shaposhnikova Representation of Quasi-Norms of Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type with Weak Reverse Doubling Property

本文提出弱反倍增条件和弱测度密度条件，建立了球准Banach函数空间的准范数表示公式，并发现当空间有限时该极限趋于零。

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón--Zygmund Operators

Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…

Square functions associated to Schrodinger operators

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrodinger operators of the form $L=-Δ+V$, where the nonnegative potential $V$ satisfies a reverse Holder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in $H^1$, $L^p$ and $BMO$ of classical $L$-square functions.

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

Let $({\mathcal X},ρ,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, a…

Sincov's inequalities on topological spaces

Assume that $X$ is a non-empty set and $T$ and $S$ are real or complex mappings defined on the product $X \times X$. Additive and multiplicative Sincov's equations are: $$T(x,z) = T(x, y ) + T(y, z)$$ and $$S(x,z) = S(x, y ) \cdot S(y, z),$$ respectively. Both equations play important roles in many areas of mathematics. In the present paper we study related inequalities. We deal with functional inequality $$ G(x,z) \leq G(x, y ) \cdot G(y, z), \quad x , y, z \in X $$ and we assume that $X$ is a topological space and $G\colon X \times X \to \mathbb{R}$ is a continuous mapping. In some our state…