In this paper, we establish a representation formula for fractional integrals. As consequence, two integral operators $I_{\lambda_1}$ and $I_{\lambda_2}$, prove Bloom type inequality \begin{align*} \mbox{\hbox to 8em{}}& \hskip -8em \left\|\big[I_{\lambda_1}^1,\big[b,I_{\lambda_2}^2\big]\big] \right\|_{L^{p_2}(L^{p_1})(\mu_2^{p_2}\times\mu_1^{p_1})\rightarrow L^{q_2}(L^{q_1})(\sigma_2^{q_2}\tim...

For any distinct two primes \(p_1\equiv p_2\equiv 3\) \((\text {mod }4)\), let \(h(-p_1)\), \(h(-p_2)\) and \(h(p_1p_2)\) be the class numbers of quadratic fields \(\mathbb {Q}(\sqrt{-p_1})\), {Q}(\sqrt{-p_2})\) {Q}(\sqrt{p_1p_2})\), respectively. Let \(\omega _{p_1p_2}:=(1+\sqrt{p_1p_2})/2\) \(\varPsi (\omega _{p_1p_2})\) Hirzebruch sum _{p_1p_2}\). We show that \(h(-p_1)h(-p_2)\equiv h(p_1p_2...

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

In this paper, we study maximal and square functions associated with bilinear Bochner–Riesz means at the critical index. particular, prove that they satisfy weighted estimates from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)\) for weights \((w_1,w_2)\in A_{\textbf{P}}\) where \(p_1,p_2>1\) \(\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}\). Also, show both operators fail to weak-type end-poi...

Abstract Let G be a locally compact unimodular group, and let $\phi $ some function of n variables on . To such , one can associate multilinear Fourier multiplier, which acts -fold product the noncommutative $L_p$ -spaces group von Neumann algebra. One may also define an associated Schur Schatten classes $S_p(L_2(G))$ We generalize well-known transference results from linear case to case. In pa...

Let [ ⋅ ] be the floor function. In this paper, we show that every sufficiently large positive integer N can represented in form $$\displaystyle N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], $$ where p1, p2, andp3 are prime numbers. We also establish an asymptotic formula for number of such representations, when do not exceed given number.

Let $p$ be an odd prime, $q=p^e$, $e \geq 1$, and $\mathbb{F} = \mathbb{F}_q$ denote the finite field of $q$ elements. $f: \mathbb{F}^2\to \mathbb{F}$ $g: \mathbb{F}^3\to functions, let $P$ $L$ two copies 3-dimensional vector space $\mathbb{F}^3$. Consider a bipartite graph $\Gamma_\mathbb{F} (f, g)$ with vertex partitions edges defined as follows: for every $(p)=(p_1,p_2,p_3)\in P$ $[l]= [l_1,...