I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^ …

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable …

$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by …

Lets define new differintegral formula as $$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ or, equivalently, $$\mathbb{D}^s_xf(x …

Hello, I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elli …

Suppose we have the following differential inequality $F''(w)\le \frac{p-1}{p}\frac{(F'(w))^2}{F(w)}$ on $w\in(w_0,w_0+\varepsilon)$, $w_0>0$, $\varepsilon>0$, $p>1$. In addition, …

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claim …

It is well known that for a given volume $V$, a sphere is the shape that minimizes the surface area. I am interested in the same problem under the constraint that the shape must li …

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions $$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu …

Hi, The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$ where subscript $y,z$ indicates derivatives and $A,B,C,D$ a …

Let $\mathbb{T}=[0,1]$ be identified with the circle $\{ e^{2 \pi it} : t \in [0,1] \}$, $\delta_0 \in M(\mathbb{T})$ be the Dirac measure at $0 \in \mathbb{T}$. Suppose $f \in L^1 …

I want to prove the following proposition: The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$. And, as I think, the same is true for the function $w\to (-Li_{p} …

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further a …

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this quest …

Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by Calderón, Lions, Peetre, et al. allow us to interpolate the continuity of two operato …