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I asked this question on math.stackexchange: Does this integral converge when $\frac{1}{p}+\frac{1}{q}\ge1$? No answers or very useful comments there. May be it is more appropraite for mathoverflow. F …

Fix $\alpha \in (0,1)$ and $\psi\in C^{\infty}_{c}(\mathbb{R}\to \mathbb{R})$. For a smooth function $\phi\geq 0$ define the integral $$J_{\alpha}(\phi):=\int \frac{\psi}{\phi^{\alpha}}.$$ If $|\phi^ …

Let $0<\alpha<1$ and define $$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$ The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of $Hf(x):=\int \frac{f(y)} …

Let $0<\alpha_{j}<1$, $j=1,\dots,d+1$. I am trying to estimate the following singular integral: $$I(y_{1},\dots,y_{d},z):=\int_{\substack{ x\in[0,1]^{d}\\ 1/2<|x|<1}} \frac{d x_{1} \dots d x_{d}}{|x_{ …

Let $k_{1},\dots, k_{d}>1$ be integers and consider the integral $$J_{\lambda }=\int_{\mathbb{S}^{d-1}}e^{-\lambda \left(x^{2k_{1}}_{1}+\dots+ x^{2k_{d}}_{d}\right)} d\sigma(x)$$ where $d\sigma$ denot …

Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions: $$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t} \int_{|x-y| …

This is a transfer of the question https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ and $ …

Let $0<\theta_1,\theta_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support. Consider the oscillatory integral $$I(t):=\int_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta_1}) (y-e^ …

Let $\alpha>1$ not necessarily an integer, and let $\psi:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function with compact support. Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\infty}\ …

Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$. I am trying to determine the asymptotic behaviour of $$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\frac{dx}{|x-a|^{\alpha}| …

Let $0<a<d/2$, let $B$ be the unit ball in $\mathbb{R}^{d}$ centered at the origin, and let $f:B \to [0,\infty[$ be a a smooth function such that (1) $f(x)\geq f(0).$ (2) $\nabla f(x)\neq 0,\quad \for …

Let $a\in (0,1)$ and define $$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$ Note that rescaling $2^{j} s\mapsto s$ shows that $$J(j)\leq 2^{-j(1+a)}\int_{0} …

I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator $$Tf(x)=\i …

Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$. I am trying to show that $$\int_{\mathbb{R}}\int_{\mathbb{R}} \,K(y,z)\, \frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,d …

My question is on Brascamp-Lieb-inequality on the Euclidean sphere (which can be viewed as an analogue of Young's inequality on the sphere) obtained in [1]. (See also this question: Brascamp-Lieb ine …